Biostatistics Quiz: Proportion Tests

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Proportion TestsQuestion 1 of 20

A manufacturer tests whether the defect rate has increased from the standard 3%. In 400 items, 18 are defective. The sample proportion is 0.045. What is the standard error for this one-proportion test?

\sqrt{\frac{0.03 \times 0.97}{400}} = 0.0085

\sqrt{\frac{0.045 \times 0.955}{400}} = 0.0104

\sqrt{\frac{0.03 \times 0.045}{400}} = 0.0055

\sqrt{0.03 \times 0.97} = 0.1697

\frac{0.045}{\sqrt{400}} = 0.0023

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Biostatistics Quiz

Biostatistics Quiz: Proportion Tests

Practice Proportion Tests in Biostatistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Proportion Tests, giving you a quick way to practice the rules, question types, and explanations that matter most for Biostatistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

$\sqrt{\frac{0.03 \times 0.97}{400}} = 0.0085$ (correct answer)
$\sqrt{\frac{0.045 \times 0.955}{400}} = 0.0104$
$\sqrt{\frac{0.03 \times 0.045}{400}} = 0.0055$
$\sqrt{0.03 \times 0.97} = 0.1697$
$\frac{0.045}{\sqrt{400}} = 0.0023$

Explanation: When you encounter a hypothesis test for a single proportion, you need to carefully distinguish between the null hypothesis value and the sample proportion when calculating the standard error. This is a crucial concept that trips up many students. For hypothesis testing of proportions, the standard error uses the null hypothesis proportion, not the sample proportion. Since the manufacturer is testing whether the defect rate has increased from the standard 3%, your null hypothesis is that

p_0 = 0.03

. The standard error formula is

SE = \sqrt{\frac{p_0(1-p_0)}{n}}

, where

p_0

is the hypothesized proportion under the null hypothesis. Choice A is correct:

\sqrt{\frac{0.03 \times 0.97}{400}} = 0.0085

. This properly uses the null hypothesis value of 0.03 and its complement 0.97, giving you the standard error needed for the test statistic. Choice B uses the sample proportion (0.045) instead of the null hypothesis proportion. This is the standard error you'd use for a confidence interval, not a hypothesis test. Choice C incorrectly multiplies the null hypothesis proportion by the sample proportion, which has no statistical meaning. Choice D forgets to divide by the sample size entirely, giving you just

\sqrt{p_0(1-p_0)}

without the

\sqrt{n}

in the denominator. Key takeaway: For hypothesis tests about proportions, always use the null hypothesis proportion in your standard error calculation. Save the sample proportion for confidence intervals.

Question 2

A pharmaceutical company tests whether their new drug has a cure rate different from the standard 40%. In a trial of 200 patients, 92 were cured. The test statistic is calculated as $z = 1.73$ . What is the p-value for this two-tailed test?

0.0418
0.0836 (correct answer)
0.9164
0.9582
1.73

Explanation: When you encounter hypothesis testing problems involving p-values, remember that the p-value represents the probability of observing your test statistic (or something more extreme) if the null hypothesis were true. For two-tailed tests, you need to account for both directions of the alternative hypothesis. With a test statistic of

z = 1.73

, you first find the area to the right of this value in the standard normal distribution. Using a z-table or calculator,

P(Z > 1.73) = 0.0418

. However, since this is a two-tailed test (testing whether the cure rate is "different from" 40%, not specifically higher or lower), you must double this probability to account for both tails:

p\text{-value} = 2 × 0.0418 = 0.0836

. Choice A (0.0418) represents the common error of calculating only the one-tailed p-value. This would be correct if the company were testing whether their drug was specifically better than the standard, but the question asks about any difference. Choice C (0.9164) gives you

1 - 0.0836

, which represents the probability of not getting a result this extreme – essentially the complement of the p-value. Choice D (0.9582) is

1 - 0.0418

, combining two errors: using the one-tailed value and then taking its complement. Study tip: Always check whether the hypothesis test is one-tailed or two-tailed by looking at the alternative hypothesis wording. Words like "different from," "not equal to," or "changed" signal two-tailed tests, requiring you to double the single-tail probability.

Question 3

A medical researcher claims that less than 25% of patients experience side effects from a new medication. In a study of 80 patients, 16 experienced side effects. What type of test should be performed?

One-tailed test with $H_a: p < 0.25$ (correct answer)
Two-tailed test with $H_a: p \neq 0.25$
One-tailed test with $H_a: p > 0.25$
Two-tailed test with $H_a: p < 0.25$
One-tailed test with $H_a: \hat{p} < 0.20$

Explanation: When you encounter a hypothesis testing question, start by carefully reading what the researcher claims and determine the direction of the test based on that claim. The researcher claims that "less than 25%" of patients experience side effects. This is a directional claim suggesting the true proportion is below 25%, not just different from 25%. In hypothesis testing, the alternative hypothesis (

H_a

) represents what the researcher is trying to prove or support with evidence. Since the claim is specifically that the proportion is less than 25%, this calls for a one-tailed test with

H_a: p < 0.25

. The null hypothesis would be

H_0: p \geq 0.25

(the proportion is 25% or higher). The sample data (16 out of 80 patients = 20% experiencing side effects) would be used to test whether there's sufficient evidence to support the claim that the true proportion is less than 25%. Option B is incorrect because a two-tailed test (

H_a: p \neq 0.25

) would be appropriate if the researcher claimed the proportion was simply "different from" 25%, without specifying a direction. Option C is wrong because

H_a: p > 0.25

tests the opposite direction—whether the proportion is greater than 25%. Option D combines the wrong test type (two-tailed) with an impossible notation, since two-tailed tests use

\neq

, not directional symbols. Study tip: Always match the alternative hypothesis to the researcher's specific claim. Words like "less than," "greater than," or "more than" signal one-tailed tests, while "different from" or "not equal to" indicate two-tailed tests.

Question 4

A company claims their customer satisfaction rate is 90%. A consultant randomly surveys 200 customers and finds 170 are satisfied. To test this claim at $\alpha = 0.01$ , what is the critical value for a two-tailed test?

$\pm 1.96$
$\pm 2.33$
$\pm 2.58$ (correct answer)
$-2.33$
$1.645$

Explanation: When you encounter hypothesis testing problems involving proportions, you need to identify the significance level and whether it's a one-tailed or two-tailed test to find the correct critical value from the standard normal distribution. This is a two-tailed test because the company claims a specific satisfaction rate (90%), and you're testing whether the actual rate differs in either direction. With

\alpha = 0.01

for a two-tailed test, you split this significance level equally between both tails: 0.005 in each tail. This means you need the z-value that leaves 0.005 in each tail, or equivalently, the z-value with 0.995 cumulative probability (since 1 - 0.005 = 0.995). Looking up 0.995 in the standard normal table gives you

z = 2.58

, making the critical values

\pm 2.58

for this two-tailed test. Choice A (

\pm 1.96

) represents the critical values for

\alpha = 0.05

in a two-tailed test, which is the most common significance level but not what's specified here. Choice B (

\pm 2.33

) corresponds to

\alpha = 0.01

in a one-tailed test, where all 0.01 goes into one tail. Choice D (

-2.33

) gives only the left critical value for a one-tailed test, ignoring that this is two-tailed. Remember this pattern: for two-tailed tests, always divide

\alpha

by 2 before finding your critical values. The common critical values to memorize are

\pm 1.96

for

\alpha = 0.05

and

\pm 2.58

for

\alpha = 0.01

in two-tailed tests.

Question 5

A political pollster wants to test if support for a candidate differs between urban and rural voters. Urban: 120 supporters out of 300 voters; Rural: 80 supporters out of 200 voters. Before conducting the test, which assumption should be verified first?

The urban and rural populations must be equal in size
Both sample proportions must be between 0.3 and 0.7
The samples must be independent and randomly selected (correct answer)
The total sample size must exceed 500 voters
The difference in sample sizes must be less than 50

Explanation: When comparing proportions between two groups, you're typically performing a two-proportion z-test. The foundation of any valid statistical test lies in meeting its underlying assumptions before examining the data or calculating test statistics. The correct answer is C because independence and random sampling form the cornerstone assumptions for proportion tests. Your samples must be independent (one group's responses don't influence the other's) and randomly selected from their respective populations. Without these conditions, your statistical inference becomes meaningless regardless of sample sizes or proportions. Random sampling ensures your samples represent their populations, while independence prevents bias from creeping into your comparison. Option A is incorrect because equal population sizes aren't required for two-proportion tests. You can validly compare groups of different sizes - urban and rural populations naturally differ in real-world scenarios. Option B misunderstands the assumption about expected counts, not sample proportions. The actual requirement is that expected frequencies in each cell should be at least 5, which relates to the normal approximation validity. The specific range 0.3-0.7 has no statistical basis. Option D incorrectly focuses on total sample size. While larger samples generally improve power, there's no magic threshold of 500. The key is having adequate expected counts in each category (typically ≥5), which depends on both sample size and proportion values. Study tip: For any hypothesis test, always verify assumptions before diving into calculations. Independence and randomness are fundamental - without them, even perfect arithmetic won't yield valid conclusions.

Question 6

Two clinics are comparing treatment success rates. Clinic A: 42 successes in 60 treatments; Clinic B: 35 successes in 50 treatments. If the p-value for testing equal success rates is 0.08, what can be concluded at $\alpha = 0.05$ ?

Reject $H_0$ ; the success rates are significantly different
Fail to reject $H_0$ ; insufficient evidence of different success rates (correct answer)
Accept $H_0$ ; the success rates are proven to be equal
Reject $H_0$ because $0.08 < 0.10$
The test is inconclusive because $0.05 < 0.08 < 0.10$

Explanation: When comparing proportions between two groups, you're testing whether observed differences could reasonably be due to chance. This requires understanding the relationship between p-values and significance levels. Here, Clinic A has a 70% success rate (42/60) while Clinic B has a 70% success rate (35/50). Despite appearing identical when rounded, the statistical test yielded p = 0.08. Since this p-value exceeds our significance level of

\alpha = 0.05

, we fail to reject the null hypothesis. This means we don't have sufficient evidence to conclude the success rates differ significantly. Answer B correctly states this conclusion: we fail to reject

H_0

due to insufficient evidence of different success rates. Answer A is wrong because rejecting

H_0

requires p ≤

\alpha

. Since 0.08 > 0.05, we cannot reject the null hypothesis. Answer C makes a critical statistical error by saying we "accept"

H_0

and that rates are "proven equal." In hypothesis testing, we never accept

H_0

or prove it true—we only fail to find sufficient evidence against it. The rates might still differ; our sample just wasn't large enough to detect the difference. Answer D uses the wrong significance level. The question specifically states

\alpha = 0.05

, not 0.10. Even if

\alpha

were 0.10, this reasoning would be backwards since 0.08 < 0.10 would mean rejecting

H_0

, not the conclusion stated. Study tip: Remember that p-value >

\alpha

always means "fail to reject

H_0

"—never "accept

H_0

" or "prove equality."

Question 7

A researcher claims that more than 60% of college students own laptops. In a sample of 150 students, 98 own laptops. What is the appropriate alternative hypothesis?

$H_a: p > 0.60$ (correct answer)
$H_a: p < 0.60$
$H_a: p \neq 0.60$
$H_a: \hat{p} > 0.653$
$H_a: p = 0.653$

Explanation: When you encounter hypothesis testing problems, the key is to carefully read what the researcher is claiming and translate that claim into the appropriate alternative hypothesis. The alternative hypothesis (

H_a

) always represents what the researcher is trying to prove or demonstrate. In this problem, the researcher claims that "more than 60% of college students own laptops." This directional claim translates directly to

H_a: p > 0.60

, where

p

represents the true population proportion of students who own laptops. The null hypothesis would be

H_0: p \leq 0.60

, representing the status quo or opposing view. Option A (

H_a: p > 0.60

) correctly captures the researcher's claim in proper hypothesis testing format. Option B (

H_a: p < 0.60

) represents the opposite claim—that fewer than 60% own laptops—which contradicts what the researcher wants to prove. Option C (

H_a: p \neq 0.60

) would be appropriate for a two-tailed test where the researcher claims the proportion is simply different from 60% (either higher or lower), but that's not what's stated here. Option D (

H_a: \hat{p} > 0.653

) makes a fundamental error by using the sample proportion (

\hat{p} = 98/150 = 0.653

) instead of the population parameter

p

, and uses a value derived from the sample data rather than the researcher's original claim. Remember: alternative hypotheses are always about population parameters (

p

\mu

), never sample statistics (

\hat{p}

\bar{x}

), and they must directly reflect the researcher's specific claim.

Question 8

A school district compares graduation rates between two high schools. School X: 180 graduates out of 200 seniors; School Y: 165 graduates out of 180 seniors. The calculated z-statistic is -1.45. What does this negative value indicate?

School X has a significantly lower graduation rate than School Y
The sample proportion for School X is less than that for School Y (correct answer)
An error was made in the calculation since z-statistics cannot be negative
The pooled proportion is negative, indicating an invalid test
School Y has insufficient sample size for a valid comparison

Explanation: When you encounter z-statistics in two-proportion hypothesis tests, remember that the sign of the z-value tells you about the direction of the difference between your sample proportions, not about statistical significance. Let's calculate the actual proportions: School X has 180/200 = 0.90 (90%) graduation rate, while School Y has 165/180 = 0.917 (91.7%) graduation rate. Since School X's proportion (0.90) is indeed lower than School Y's proportion (0.917), the z-statistic correctly reflects this relationship with its negative value. The z-statistic formula is

z = \frac{(\hat{p_1} - \hat{p_2}) - 0}{\text{standard error}}

, where

\hat{p_1}

represents School X's proportion. Since 0.90 - 0.917 = -0.017, the numerator is negative, producing a negative z-value. Choice A is incorrect because statistical significance depends on whether |z| exceeds the critical value (typically 1.96 for α = 0.05). Here, |-1.45| < 1.96, so there's no significant difference. Choice C reflects a fundamental misunderstanding—z-statistics can absolutely be negative and simply indicate which sample proportion is smaller. Choice D is nonsensical because pooled proportions in two-sample tests are always positive values between 0 and 1, calculated as the total successes divided by total sample size. Study tip: The sign of your z-statistic always matches the sign of (proportion₁ - proportion₂). Negative z-values are completely normal and simply tell you the direction of the difference—don't confuse the sign with significance testing.

Question 9

A marketing firm claims that 35% of consumers prefer their product over competitors. A survey of 250 consumers finds 75 prefer their product. The test statistic is $z = -1.68$ . At $\alpha = 0.10$ for a two-tailed test, what is the conclusion?

Reject $H_0$ because $|-1.68| > 1.645$ (correct answer)
Fail to reject $H_0$ because $|-1.68| < 1.96$
Reject $H_0$ because $-1.68 < 0$
Fail to reject $H_0$ because $1.68 > 1.645$
Inconclusive because the test statistic is negative

Explanation: This question tests your understanding of hypothesis testing with z-statistics, specifically how to make decisions using critical values in two-tailed tests. When you have a two-tailed test at

\alpha = 0.10

, you split the significance level equally between both tails: 0.05 in each tail. This means your critical z-values are

\pm 1.645

(the z-score that leaves 0.05 in each tail). You reject

H_0

if your test statistic falls in either rejection region:

z < -1.645

z > 1.645

. Equivalently, you can check if

|z| > 1.645

. With

z = -1.68

, we have

|-1.68| = 1.68 > 1.645

, so we reject

H_0

. The marketing firm's claim of 35% preference is not supported by the sample data. Answer A is correct because it properly compares the absolute value of the test statistic (1.68) to the correct critical value (1.645) for

\alpha = 0.10

. Answer B incorrectly uses 1.96, which is the critical value for

\alpha = 0.05

, not 0.10. This is a common mix-up between significance levels. Answer C gives the right conclusion but wrong reasoning. The sign of z doesn't determine rejection by itself—it's whether the absolute value exceeds the critical value that matters. Answer D incorrectly suggests we fail to reject, even though 1.68 > 1.645 actually supports rejection. Study tip: Always match your critical value to the given significance level. For two-tailed tests, remember that common critical values are 1.645 (

\alpha = 0.10

) and 1.96 (

\alpha = 0.05

Question 10

A pharmaceutical company tests whether men and women respond differently to a new drug. Men: 45 positive responses out of 80 participants; Women: 38 positive responses out of 70 participants. What assumption is most critical for the validity of this test?

The sample sizes for men and women must be exactly equal
Both groups must have the same underlying response rate
The participants in each group must be independent of each other (correct answer)
The total sample size must be at least 100 participants
The response rates must both be between 40% and 60%

Explanation: When comparing proportions between two groups (like response rates between men and women), you're typically conducting a two-proportion z-test or chi-square test. The validity of these statistical tests hinges on several key assumptions about your data structure. The most fundamental assumption is that observations within each group must be independent of each other (C). This means one person's response cannot influence another person's response. If participants discussed the treatment, were family members, or were treated in groups where they could influence each other, this independence would be violated and your statistical test results would be unreliable. Independence ensures that each observation provides unique information about the treatment effect. Looking at why the other options are incorrect: (A) is wrong because equal sample sizes, while sometimes beneficial for power, are not required for valid statistical comparison between groups. You can legitimately compare groups of different sizes. (B) misunderstands the purpose of the test—you're actually testing whether the groups have different response rates, so assuming they're the same would defeat the point. (D) is incorrect because while larger samples generally provide more power, there's no universal minimum of 100 participants required for validity. Study tip: For any statistical test comparing groups, always ask yourself about the independence assumption first. Questions about "validity" in biostatistics often focus on whether the fundamental assumptions of independence, randomness, or appropriate sampling have been met, rather than technical details about sample sizes or calculation methods.

Question 11

A quality assurance manager tests whether the defect rate has changed from the historical 2%. In 600 items, 18 are defective. The sample proportion is 0.03. What is the test statistic?

$z = \frac{0.03 - 0.02}{\sqrt{\frac{0.02 \times 0.98}{600}}} = 1.75$ (correct answer)
$z = \frac{0.03 - 0.02}{\sqrt{\frac{0.03 \times 0.97}{600}}} = 1.44$
$z = \frac{18 - 12}{\sqrt{600 \times 0.02 \times 0.98}} = 1.73$
$z = \frac{0.03}{\sqrt{\frac{0.02 \times 0.98}{600}}} = 8.31$
$z = \frac{0.03 - 0.02}{0.02} = 0.50$

Explanation: When testing a population proportion, you're comparing your sample proportion to a hypothesized value using a z-test. The key is identifying which proportion goes in the denominator of the standard error formula. For hypothesis tests about proportions, the test statistic formula is

z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}

, where

\hat{p}

is your sample proportion,

p_0

is the hypothesized proportion, and

n

is sample size. Crucially, you use the hypothesized proportion (

p_0 = 0.02

) in the standard error, not the sample proportion. Answer A correctly applies this formula:

z = \frac{0.03 - 0.02}{\sqrt{\frac{0.02 \times 0.98}{600}}} = 1.75

. The denominator uses the historical rate of 0.02 because we're testing whether the true rate has changed from this value. Answer B makes a common error by using the sample proportion (0.03) in the standard error calculation instead of the hypothesized proportion. This is incorrect for hypothesis testing. Answer C uses the raw counts (18 defective items vs. expected 12) rather than proportions. While mathematically this can work, it doesn't match the standard hypothesis testing format and the calculation shown gives 1.73, not the correct 1.75. Answer D incorrectly omits the hypothesized proportion from the numerator, testing whether the sample proportion equals zero rather than whether it differs from 0.02. Remember: In proportion hypothesis tests, always use the null hypothesis value (

p_0

) in your standard error calculation, never the sample proportion.

Question 12

An epidemiologist compares infection rates between two cities during an outbreak. City A: 65 infections out of 1000 residents tested; City B: 48 infections out of 800 residents tested. If the p-value is 0.12, what can be concluded at $\alpha = 0.05$ ?

The infection rates are significantly different between cities
City A has a significantly higher infection rate than City B
There is insufficient evidence that infection rates differ between cities (correct answer)
The infection rates are proven to be equal between cities
The test should be repeated with a larger significance level

Explanation: When you encounter a hypothesis testing question in biostatistics, focus on the relationship between the p-value and your significance level (

\alpha

) to determine what the data tells you about the research question. Here, you're comparing infection rates: City A has 65/1000 = 6.5% and City B has 48/800 = 6.0%. The epidemiologist tested whether these rates are significantly different, yielding a p-value of 0.12. Since p = 0.12 >

\alpha

= 0.05, you fail to reject the null hypothesis. In hypothesis testing, this means there's insufficient evidence to conclude the infection rates differ between cities. The observed difference could reasonably be due to random variation rather than a true difference between populations. Answer C correctly states this conclusion. Answer A is wrong because "significantly different" requires p ≤

\alpha

, which we don't have. Answer B makes two errors: it assumes significance when there isn't any, and it implies a one-tailed test focusing on City A being higher, when the question suggests a two-tailed comparison. Answer D commits a classic statistical error—failing to reject the null hypothesis never "proves" equality. We simply lack sufficient evidence to detect a difference; a true difference might exist but be too small for our sample sizes to detect. Study tip: Remember that p >

\alpha

means "insufficient evidence," never "proof of no difference." Statistical tests can only provide evidence against the null hypothesis, not prove it true.

Question 13

A health inspector tests whether the contamination rate at a food processing plant exceeds 5%. In 300 samples, 21 show contamination. If $\alpha = 0.05$ and this is a one-tailed test, what is the critical value?

$1.96$
$1.645$ (correct answer)
$2.33$
$-1.645$
$\pm 1.645$

Explanation: When you encounter hypothesis testing problems involving critical values, you need to identify three key elements: the significance level (

\alpha

), whether it's a one-tailed or two-tailed test, and the appropriate distribution to use. This problem involves testing whether a contamination rate exceeds 5%, which is a one-tailed test since we're only concerned with rates greater than 5%. With a sample size of 300, we can use the standard normal distribution (z-distribution) because the sample is large enough for the Central Limit Theorem to apply. For a one-tailed test with

\alpha = 0.05

, you need the z-value that leaves 5% in the upper tail of the standard normal distribution. This critical value is

z = 1.645

, making B the correct answer. Let's examine why the other options are wrong: A)

1.96

is the critical value for a two-tailed test with

\alpha = 0.05

, where you split the significance level between both tails (2.5% each). C)

2.33

would be the critical value for a one-tailed test with

\alpha = 0.01

, which is more stringent than our given significance level. D)

-1.645

would be correct for a left-tailed test (testing if the rate is significantly less than 5%), but this problem asks whether the rate exceeds 5%, making it a right-tailed test. Study tip: Remember that one-tailed tests use

1.645

for

\alpha = 0.05

and two-tailed tests use

1.96

. The sign depends on which tail you're testing—positive for right-tailed, negative for left-tailed.

Question 14

A researcher claims that 65% of college students prefer online learning. In a sample of 120 students, 72 prefer online learning. What is the appropriate null hypothesis for testing this claim?

$H_0: \hat{p} = 0.65$
$H_0: p = 0.60$
$H_0: p = 0.65$ (correct answer)
$H_0: \hat{p} = 0.60$
$H_0: p \neq 0.65$

Explanation: When you encounter hypothesis testing questions, remember that the null hypothesis (

H_0

) always represents the status quo or the claim being tested. It uses the population parameter, not sample statistics. In this problem, the researcher claims that 65% of college students prefer online learning. This claim becomes our null hypothesis, so we write

H_0: p = 0.65

, where

p

represents the true population proportion. The sample data (72 out of 120 students, or 60%) will be used to test whether this claim is reasonable. Let's examine why the other options are incorrect: Option A (

H_0: \hat{p} = 0.65

) uses

\hat{p}

, which represents the sample proportion. Null hypotheses are always about population parameters, never sample statistics. This is a fundamental error in hypothesis testing. Option B (

H_0: p = 0.60

) uses the correct notation but the wrong value. While 0.60 is the observed sample proportion (72/120), the null hypothesis should reflect the researcher's claim (65%), not what we observed in our sample. Option D (

H_0: \hat{p} = 0.60

) combines both errors: it uses sample notation (

\hat{p}

) and bases the hypothesis on the sample result rather than the original claim. Study tip: Always distinguish between population parameters (

p

\mu

\sigma

) and sample statistics (

\hat{p}

\bar{x}

s

). The null hypothesis tests claims about populations, so it always uses parameter notation. The researcher's original claim, not your sample results, determines the null hypothesis value.

Question 15

A quality control manager tests whether the defect rate has changed from the historical 8%. In 500 items, 50 are defective. If the calculated test statistic is $z = 1.47$ , what conclusion should be drawn at $\alpha = 0.05$ ?

Reject $H_0$ because $1.47 > 0.05$
Fail to reject $H_0$ because $|1.47| < 1.96$ (correct answer)
Reject $H_0$ because $1.47 > 1.645$
Fail to reject $H_0$ because the sample proportion equals 0.10
Reject $H_0$ because $1.47^2 > 1.96$

Explanation: When you encounter hypothesis testing problems involving proportions, you need to identify the test type and compare your test statistic to the appropriate critical value. This is a two-tailed test since we're checking whether the defect rate has "changed" (could be higher or lower) from the historical 8%. For a two-tailed z-test at

\alpha = 0.05

, the critical values are

\pm 1.96

. Since our calculated test statistic is

z = 1.47

, we have

|1.47| = 1.47 < 1.96

. Because the test statistic falls within the acceptance region (between -1.96 and +1.96), we fail to reject the null hypothesis. There isn't sufficient evidence that the defect rate has changed significantly from 8%. Option A incorrectly compares the test statistic directly to the significance level (1.47 > 0.05), which makes no statistical sense—you never compare z-values to alpha values directly. Option C treats this as a one-tailed test by using the critical value 1.645, but the phrase "has changed" indicates we're testing for difference in either direction, requiring a two-tailed approach. Option D mentions the sample proportion (50/500 = 0.10) but doesn't properly connect this to the hypothesis testing framework—the decision depends on the test statistic relative to critical values, not just the sample proportion value. Remember: "changed" or "different" signals a two-tailed test with critical values

\pm 1.96

\alpha = 0.05

, while "increased" or "decreased" would indicate one-tailed tests with

\pm 1.645

Question 16

Two laboratories are comparing error rates in blood tests. Lab A: 8 errors in 400 tests; Lab B: 15 errors in 500 tests. To perform a two-proportion z-test, what must be calculated first?

The difference in sample sizes (500 - 400 = 100)
The individual sample proportions (0.02 and 0.03)
The pooled proportion $\hat{p} = \frac{8 + 15}{400 + 500}$ (correct answer)
The average of the two sample proportions $\frac{0.02 + 0.03}{2}$
The ratio of sample sizes $\frac{500}{400} = 1.25$

Explanation: When comparing proportions between two groups, you're testing whether the difference between them is statistically significant. The two-proportion z-test requires a specific approach to calculate the test statistic properly. The correct answer is C because the two-proportion z-test uses a pooled proportion under the null hypothesis assumption that both populations have the same true proportion. This pooled proportion

\hat{p} = \frac{8 + 15}{400 + 500} = \frac{23}{900} = 0.0256

combines all successes (errors) over all trials from both labs. This pooled value is then used to calculate the standard error for the test statistic. Option A is wrong because while you need both sample sizes for the calculation, simply finding their difference (100) serves no purpose in the two-proportion z-test formula. Option B represents an intermediate step - you do calculate the individual sample proportions (Lab A: 8/400 = 0.02; Lab B: 15/500 = 0.03), but these alone aren't sufficient to perform the test. The pooled proportion must be calculated first to proceed with the hypothesis test. Option D incorrectly suggests averaging the two sample proportions, which gives

\frac{0.02 + 0.03}{2} = 0.025

. This average doesn't account for the different sample sizes and isn't the proper way to pool proportions in hypothesis testing. Remember: Two-proportion z-tests always require pooling under the null hypothesis. Look for the formula that combines total successes over total trials from both groups - this distinguishes it from confidence intervals, which use individual sample proportions.

Question 17

A restaurant owner claims that more than 80% of customers are satisfied with service. A survey of 120 customers finds 102 are satisfied. To test this claim at $\alpha = 0.01$ , what decision should be made if the calculated test statistic is $z = 2.19$ ?

Reject $H_0$ because $2.19 > 2.33$
Fail to reject $H_0$ because $2.19 < 2.33$ (correct answer)
Reject $H_0$ because $2.19 > 1.96$
Fail to reject $H_0$ because this is a one-tailed test
Reject $H_0$ because the sample proportion exceeds 0.80

Explanation: When you encounter a hypothesis testing question, you need to systematically compare your test statistic to the critical value based on your significance level and test type. This is a one-tailed test since the claim is "more than 80%" (testing

H_0: p \leq 0.8

H_1: p > 0.8

). With

\alpha = 0.01

for a right-tailed test, the critical value is

z_{0.01} = 2.33

. Since your calculated test statistic

z = 2.19

is less than 2.33, you fail to reject

H_0

. The evidence isn't strong enough at the 1% significance level to support the owner's claim. Option A incorrectly states that

2.19 > 2.33

, which is mathematically false. Option C uses the wrong critical value (

z = 1.96

corresponds to

\alpha = 0.05

, not 0.01) and would lead to an incorrect rejection. Option D gives the right conclusion but for the wrong reason—being a one-tailed test doesn't automatically mean you fail to reject

H_0

; the decision depends on comparing your test statistic to the appropriate critical value. The correct answer is B because

2.19 < 2.33

, meaning the test statistic falls in the non-rejection region. Study tip: Always identify three things before making your decision: (1) Is this one-tailed or two-tailed? (2) What's the correct critical value for your

\alpha

? (3) Does your test statistic exceed that critical value? Double-check your critical values—confusing 0.01 and 0.05 significance levels is a common mistake.

Question 18

Two hospitals are comparing infection rates. Hospital A had 18 infections in 150 surgeries, while Hospital B had 12 infections in 180 surgeries. Which condition must be verified before performing a two-proportion z-test?

Both sample sizes must be greater than 30 patients each
The pooled sample proportion must equal 0.5 exactly
Each hospital must have at least 5 successes and 5 failures (correct answer)
The difference in sample proportions must be normally distributed
Both populations must have identical infection rates

Explanation: When performing a two-proportion z-test, you're comparing infection rates between two groups using a normal approximation to the binomial distribution. The key question is: when is this normal approximation valid? The correct answer is C. For the normal approximation to work properly, you need adequate "successes" and "failures" in your sample. Specifically, each group should have at least 5 infections (successes) and at least 5 non-infections (failures). Let's check: Hospital A has 18 infections and 132 non-infections (150-18), while Hospital B has 12 infections and 168 non-infections (180-12). Both hospitals satisfy this condition. Here's why the other options are incorrect: Option A confuses the two-proportion test with the one-sample t-test requirement. While sample size matters, the critical threshold isn't simply "30 patients" – it's about having enough successes and failures for the normal approximation. Option B is completely wrong; there's no requirement for the pooled proportion to equal 0.5. The pooled proportion is calculated from your actual data:

\frac{18+12}{150+180} = 0.091

. Option D puts the cart before the horse – we use this condition to ensure the sampling distribution will be approximately normal, not verify it afterward. Study tip: For any proportion test, immediately check the "5 and 5 rule" – at least 5 successes and 5 failures in each group. This is the gateway condition that determines whether you can proceed with the z-test.

Question 19

An educator tests whether online students have a different completion rate than in-person students. Online: 85 completions out of 120 students; In-person: 78 completions out of 100 students. The standard error for the difference in proportions is approximately 0.062. What is the test statistic?

$z = \frac{0.708 - 0.780}{0.062} = -1.16$ (correct answer)
$z = \frac{0.780 - 0.708}{0.062} = 1.16$
$z = \frac{85 - 78}{0.062} = 112.90$
$z = \frac{0.708 + 0.780}{0.062} = 24.00$
$z = \frac{|0.708 - 0.780|}{0.062^2} = 18.75$

Explanation: When testing for differences between two proportions, you're looking at whether two groups have significantly different rates of some outcome. The key is setting up your test statistic correctly using the formula

z = \frac{\hat{p_1} - \hat{p_2}}{SE}

, where you compare the difference in sample proportions to the standard error. First, calculate the proportions: online students have

\hat{p_1} = \frac{85}{120} = 0.708

and in-person students have

\hat{p_2} = \frac{78}{100} = 0.780

. Using the given standard error of 0.062, the test statistic becomes

z = \frac{0.708 - 0.780}{0.062} = \frac{-0.072}{0.062} = -1.16

. Answer A correctly applies this formula with the proper order (online minus in-person). Answer B makes the common error of switching the order to get a positive result, calculating

(0.780 - 0.708)/0.062 = 1.16

. While the magnitude is correct, the sign matters for interpretation—it tells you which group has the higher rate. Answer C incorrectly uses raw counts (85 - 78 = 7) instead of proportions, yielding an unrealistically large test statistic of 112.90. Answer D nonsensically adds the proportions instead of finding their difference, giving 24.00. The negative z-score in the correct answer tells you that online students have a lower completion rate than in-person students. Remember: always use proportions (not raw counts) when comparing rates between groups, and pay attention to which group you subtract from which—the sign of your test statistic provides meaningful information about the direction of the difference.

Question 20

A researcher wants to test if the proportion of students who study abroad differs between two universities. University X: 45 out of 300 students; University Y: 38 out of 250 students. What is the pooled proportion for this test?

$\frac{45 + 38}{300 + 250} = 0.151$ (correct answer)
$\frac{45}{300} + \frac{38}{250} = 0.302$
$\frac{45 - 38}{300 + 250} = 0.013$
$\frac{45}{300} - \frac{38}{250} = -0.002$
$\frac{300 + 250}{45 + 38} = 6.627$

Explanation: When comparing proportions between two groups, you need to understand the concept of pooled proportion, which estimates the overall proportion assuming no difference between groups. The pooled proportion combines all successes from both groups and divides by the total sample size from both groups. For this study abroad comparison, you take the total number of students who studied abroad (45 + 38 = 83) and divide by the total number of students surveyed (300 + 250 = 550). This gives you

\frac{83}{550} = 0.151

, which matches answer choice A. Let's examine why the other options are incorrect. Answer B adds the individual proportions (

\frac{45}{300} + \frac{38}{250} = 0.302

), but this doesn't account for the different sample sizes between universities and isn't mathematically meaningful for hypothesis testing. Answer C calculates the difference in the number of students who studied abroad divided by total sample size (

\frac{7}{550} = 0.013

), which has no statistical interpretation. Answer D finds the difference between the two individual proportions (

0.15 - 0.152 = -0.002

), which would be useful for calculating the test statistic but isn't the pooled proportion. The pooled proportion serves as your best estimate of the true proportion under the null hypothesis that both universities have the same study abroad rate. Remember this pattern: for two-proportion tests, always pool by adding numerators and adding denominators separately—never add the calculated proportions themselves.