Significance Logic and Establishing Hypotheses - AP Statistics
Card 1 of 96
A researcher stated that 
percent of Americans in the 2008 presidential election were liberal. The researcher wants to find evidence that the percent of liberal Americans was different for the 2012 presidential election. Which of the hypotheses best describes the question that the researcher is trying to answer?
A researcher stated that
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The question states that the researcher thinks the percent of liberal Americans in 2012 is different than the 
percent in 2008. The hypothesis is 2 directional. Therefore the null hypothesis is 
and the alternative is 
.
The question states that the researcher thinks the percent of liberal Americans in 2012 is different than the
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The department of labor reports that umemployment is down to 5% in the United States. A politician claims that the data was manipulated to show better numbers. The politician believes our country is worse off that unemployment must be worse than the claim made by the department of labor. What is the alternative hypothesis for a test for this situation?
The department of labor reports that umemployment is down to 5% in the United States. A politician claims that the data was manipulated to show better numbers. The politician believes our country is worse off that unemployment must be worse than the claim made by the department of labor. What is the alternative hypothesis for a test for this situation?
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The alternative hypthesis is denoted by Ha. The alternative represents the value that is being tested for and not the original claim. The original claim for the problem is unemployment is 
%. The test claim is that it is greater than 
%. This means the alternative must be 
.
The alternative hypthesis is denoted by Ha. The alternative represents the value that is being tested for and not the original claim. The original claim for the problem is unemployment is
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A study would like to determine whether meditation helps students increase focus time. They used a control group of 30 students and compared their focus time to a group of 30 meditating students and compared their average time spent meditating. What is the appropriate alternative hypothesis for this study?
A study would like to determine whether meditation helps students increase focus time. They used a control group of 30 students and compared their focus time to a group of 30 meditating students and compared their average time spent meditating. What is the appropriate alternative hypothesis for this study?
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Because we are comparing two samples, the hypothesis takes the form of 
. The study is testing the claim that meditation helps increase focus time, so this will be our alternative hypothesis. Our alternative hypothesis in words is that the difference between control and meditation groups is less than zero. That is because we think average time for meditators will be greater than the control group’s average.
Because we are comparing two samples, the hypothesis takes the form of
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A study would like to determine whether meditation helps students improve focus time. They know that the average focus time of an American 4th grader is 23 minutes. They then recruit 50 meditators and calculate their average focus time. What is the appropriate alternative hypothesis for this study?
A study would like to determine whether meditation helps students improve focus time. They know that the average focus time of an American 4th grader is 23 minutes. They then recruit 50 meditators and calculate their average focus time. What is the appropriate alternative hypothesis for this study?
Tap to reveal answer
Because we are comparing one sample to a known mean, the hypothesis takes the form of 
compared to the known value. We would like to test the claim that meditating will increase focus time, so this will be the alternative hypothesis. Therefore, the alternative that 
is greater than the average of 23 minutes.
Because we are comparing one sample to a known mean, the hypothesis takes the form of
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A researcher wants to determine whether there is a significant linear relationship between time spent meditating and time spent studying. What is the appropriate alternative hypothesis for this study?
A researcher wants to determine whether there is a significant linear relationship between time spent meditating and time spent studying. What is the appropriate alternative hypothesis for this study?
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This question is about a linear regression between time spent meditating and time spent studying. Therefore, the hypothesis is regarding Beta1, the slope of the line. We are testing a non-directional or bi-directional claim that the relationship is significant. A slope of zero indicates no relationship between the two variables. Therefore, the alternative hypothesis is that the slope is not equal to zero.
This question is about a linear regression between time spent meditating and time spent studying. Therefore, the hypothesis is regarding Beta1, the slope of the line. We are testing a non-directional or bi-directional claim that the relationship is significant. A slope of zero indicates no relationship between the two variables. Therefore, the alternative hypothesis is that the slope is not equal to zero.
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A researcher wants to compare 3 different treatments to see if any of the treatments affects study time. The three treatments studied are control group, a group given vitamins, and a group given a placebo. They found that the average time spent studying with control students was 2 hours, with students given vitamins it was 3 hours, and with placebos students studied 5 hours. Which of the following is the correct alternative hypothesis?
A researcher wants to compare 3 different treatments to see if any of the treatments affects study time. The three treatments studied are control group, a group given vitamins, and a group given a placebo. They found that the average time spent studying with control students was 2 hours, with students given vitamins it was 3 hours, and with placebos students studied 5 hours. Which of the following is the correct alternative hypothesis?
Tap to reveal answer
Because we are comparing more than 2 groups, we must use an ANOVA for this problem. For an ANOVA problem, the null hypothesis is that not all of the groups’ means are the same. 
is not correct because this hypothesis implies that all of the groups are different, which is not actually tested by the basic ANOVA test.
Because we are comparing more than 2 groups, we must use an ANOVA for this problem. For an ANOVA problem, the null hypothesis is that not all of the groups’ means are the same.
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A researcher wants to investigate the claim that taking vitamins will help a student study longer. First, the researcher collects 32 students who do not take vitamins and determines their time spent studying. Then, the 32 students are given a vitamin for 1 week. After 1 week of taking vitamins, students are again tested to determine their time spent studying. Which of the following is the correct alternative hypothesis?
A researcher wants to investigate the claim that taking vitamins will help a student study longer. First, the researcher collects 32 students who do not take vitamins and determines their time spent studying. Then, the 32 students are given a vitamin for 1 week. After 1 week of taking vitamins, students are again tested to determine their time spent studying. Which of the following is the correct alternative hypothesis?
Tap to reveal answer
Because the same students are tested twice, this is a paired study, therefore we must use a hypothesis appropriate for a paired t-test. The hypothesis for a paired t-test regards the average of the differences between before and after treatment, called 
. We are testing the claim that vitamins increase study time, therefore this will be the alternative hypothesis. This means that the difference between control – treatment is greater than zero, since the average study time for vitamin users would be greater than that of the control.
Because the same students are tested twice, this is a paired study, therefore we must use a hypothesis appropriate for a paired t-test. The hypothesis for a paired t-test regards the average of the differences between before and after treatment, called
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A researcher stated that percent of Americans in the 2008 presidential election were liberal. The researcher wants to find evidence that the percent of liberal Americans was different for the 2012 presidential election. Which of the hypotheses best describes the question that the researcher is trying to answer?
A researcher stated that
Tap to reveal answer
The question states that the researcher thinks the percent of liberal Americans in 2012 is different than the percent in 2008. The hypothesis is 2 directional. Therefore the null hypothesis is and the alternative is .
The question states that the researcher thinks the percent of liberal Americans in 2012 is different than the
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The department of labor reports that umemployment is down to 5% in the United States. A politician claims that the data was manipulated to show better numbers. The politician believes our country is worse off that unemployment must be worse than the claim made by the department of labor. What is the alternative hypothesis for a test for this situation?
The department of labor reports that umemployment is down to 5% in the United States. A politician claims that the data was manipulated to show better numbers. The politician believes our country is worse off that unemployment must be worse than the claim made by the department of labor. What is the alternative hypothesis for a test for this situation?
Tap to reveal answer
The alternative hypthesis is denoted by Ha. The alternative represents the value that is being tested for and not the original claim. The original claim for the problem is unemployment is %. The test claim is that it is greater than %. This means the alternative must be .
The alternative hypthesis is denoted by Ha. The alternative represents the value that is being tested for and not the original claim. The original claim for the problem is unemployment is
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A study would like to determine whether meditation helps students increase focus time. They used a control group of 30 students and compared their focus time to a group of 30 meditating students and compared their average time spent meditating. What is the appropriate alternative hypothesis for this study?
A study would like to determine whether meditation helps students increase focus time. They used a control group of 30 students and compared their focus time to a group of 30 meditating students and compared their average time spent meditating. What is the appropriate alternative hypothesis for this study?
Tap to reveal answer
Because we are comparing two samples, the hypothesis takes the form of . The study is testing the claim that meditation helps increase focus time, so this will be our alternative hypothesis. Our alternative hypothesis in words is that the difference between control and meditation groups is less than zero. That is because we think average time for meditators will be greater than the control group’s average.
Because we are comparing two samples, the hypothesis takes the form of
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A study would like to determine whether meditation helps students improve focus time. They know that the average focus time of an American 4th grader is 23 minutes. They then recruit 50 meditators and calculate their average focus time. What is the appropriate alternative hypothesis for this study?
A study would like to determine whether meditation helps students improve focus time. They know that the average focus time of an American 4th grader is 23 minutes. They then recruit 50 meditators and calculate their average focus time. What is the appropriate alternative hypothesis for this study?
Tap to reveal answer
Because we are comparing one sample to a known mean, the hypothesis takes the form of compared to the known value. We would like to test the claim that meditating will increase focus time, so this will be the alternative hypothesis. Therefore, the alternative that is greater than the average of 23 minutes.
Because we are comparing one sample to a known mean, the hypothesis takes the form of
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A researcher wants to determine whether there is a significant linear relationship between time spent meditating and time spent studying. What is the appropriate alternative hypothesis for this study?
A researcher wants to determine whether there is a significant linear relationship between time spent meditating and time spent studying. What is the appropriate alternative hypothesis for this study?
Tap to reveal answer
This question is about a linear regression between time spent meditating and time spent studying. Therefore, the hypothesis is regarding Beta1, the slope of the line. We are testing a non-directional or bi-directional claim that the relationship is significant. A slope of zero indicates no relationship between the two variables. Therefore, the alternative hypothesis is that the slope is not equal to zero.
This question is about a linear regression between time spent meditating and time spent studying. Therefore, the hypothesis is regarding Beta1, the slope of the line. We are testing a non-directional or bi-directional claim that the relationship is significant. A slope of zero indicates no relationship between the two variables. Therefore, the alternative hypothesis is that the slope is not equal to zero.
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A researcher wants to compare 3 different treatments to see if any of the treatments affects study time. The three treatments studied are control group, a group given vitamins, and a group given a placebo. They found that the average time spent studying with control students was 2 hours, with students given vitamins it was 3 hours, and with placebos students studied 5 hours. Which of the following is the correct alternative hypothesis?
A researcher wants to compare 3 different treatments to see if any of the treatments affects study time. The three treatments studied are control group, a group given vitamins, and a group given a placebo. They found that the average time spent studying with control students was 2 hours, with students given vitamins it was 3 hours, and with placebos students studied 5 hours. Which of the following is the correct alternative hypothesis?
Tap to reveal answer
Because we are comparing more than 2 groups, we must use an ANOVA for this problem. For an ANOVA problem, the null hypothesis is that not all of the groups’ means are the same. is not correct because this hypothesis implies that all of the groups are different, which is not actually tested by the basic ANOVA test.
Because we are comparing more than 2 groups, we must use an ANOVA for this problem. For an ANOVA problem, the null hypothesis is that not all of the groups’ means are the same.
← Didn't Know|Knew It →
A researcher wants to investigate the claim that taking vitamins will help a student study longer. First, the researcher collects 32 students who do not take vitamins and determines their time spent studying. Then, the 32 students are given a vitamin for 1 week. After 1 week of taking vitamins, students are again tested to determine their time spent studying. Which of the following is the correct alternative hypothesis?
A researcher wants to investigate the claim that taking vitamins will help a student study longer. First, the researcher collects 32 students who do not take vitamins and determines their time spent studying. Then, the 32 students are given a vitamin for 1 week. After 1 week of taking vitamins, students are again tested to determine their time spent studying. Which of the following is the correct alternative hypothesis?
Tap to reveal answer
Because the same students are tested twice, this is a paired study, therefore we must use a hypothesis appropriate for a paired t-test. The hypothesis for a paired t-test regards the average of the differences between before and after treatment, called . We are testing the claim that vitamins increase study time, therefore this will be the alternative hypothesis. This means that the difference between control – treatment is greater than zero, since the average study time for vitamin users would be greater than that of the control.
Because the same students are tested twice, this is a paired study, therefore we must use a hypothesis appropriate for a paired t-test. The hypothesis for a paired t-test regards the average of the differences between before and after treatment, called
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A pretzel company advertises that their pretzels contain less than 1.0g of of sodium per serving. You take a simple random sample of 10 pretzel servings, and calculate that the mean amount of sodium is 1.20 g, with a standard deviation of 0.1 g.
At the 95% confidence level, does your sample suggest that the pretzels actually have higher than 1.0g of sodium per serving?
A pretzel company advertises that their pretzels contain less than 1.0g of of sodium per serving. You take a simple random sample of 10 pretzel servings, and calculate that the mean amount of sodium is 1.20 g, with a standard deviation of 0.1 g.
At the 95% confidence level, does your sample suggest that the pretzels actually have higher than 1.0g of sodium per serving?
Tap to reveal answer
This is a one- tailed t-test. It is one-tailed because the question asks whether the pretzel's mean is actually higher, so we are only interested in the right hand tail. We will be using the t-distribution because the population standard deviation is not known.
First we write our hypotheses:
Now we need the appropriate formula for a t-test. We will be using standard error because we are working with the standard deviation of a sampling distribution.
Now we fill in the values from our problem
Now we must look up the t-critical value, or use technology to find the p-value.
We must find the t-critcal value by finding 
Because our test statistic 6.32 is more extreme than our critical value, we reject our null hypothesis and conclude that the pretzels do have a higher mean than 1.0.
If you calculated a p-value using technology, p=0.00006884.
Because 
, we reject our null hypothesis and conclude that the pretzels do have a higher mean than 1.0 g.
This is a one- tailed t-test. It is one-tailed because the question asks whether the pretzel's mean is actually higher, so we are only interested in the right hand tail. We will be using the t-distribution because the population standard deviation is not known.
First we write our hypotheses:
Now we need the appropriate formula for a t-test. We will be using standard error because we are working with the standard deviation of a sampling distribution.
Now we fill in the values from our problem
Now we must look up the t-critical value, or use technology to find the p-value.
We must find the t-critcal value by finding
Because our test statistic 6.32 is more extreme than our critical value, we reject our null hypothesis and conclude that the pretzels do have a higher mean than 1.0.
If you calculated a p-value using technology, p=0.00006884.
Because
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James goes to UCLA, and he believes that the atheletes of UCLA are better runners, than the country average. He did a bit of a research and found that the national average time for a two-mile run for college atheletes is 
min with a standard deviation of 
minute. He then sampled 
UCLA atheletes and found that their average two-mile time was 
minutes.
Is James' data statistically significant? Can we confirm that UCLA atheletes are better than average runners? And if so, to which level of certainty: 
, 
, 
, 
James goes to UCLA, and he believes that the atheletes of UCLA are better runners, than the country average. He did a bit of a research and found that the national average time for a two-mile run for college atheletes is
Is James' data statistically significant? Can we confirm that UCLA atheletes are better than average runners? And if so, to which level of certainty:
Tap to reveal answer
Using a Z-test (we have population SD, not sample SD) and a population of 
, we arrive at a P-value of 
, which is lower than 
, but above 
.
Using a Z-test (we have population SD, not sample SD) and a population of
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A pretzel company advertises that their pretzels contain less than 1.0g of of sodium per serving. You take a simple random sample of 10 pretzel servings, and calculate that the mean amount of sodium is 1.20 g, with a standard deviation of 0.1 g.
At the 95% confidence level, does your sample suggest that the pretzels actually have higher than 1.0g of sodium per serving?
A pretzel company advertises that their pretzels contain less than 1.0g of of sodium per serving. You take a simple random sample of 10 pretzel servings, and calculate that the mean amount of sodium is 1.20 g, with a standard deviation of 0.1 g.
At the 95% confidence level, does your sample suggest that the pretzels actually have higher than 1.0g of sodium per serving?
Tap to reveal answer
This is a one- tailed t-test. It is one-tailed because the question asks whether the pretzel's mean is actually higher, so we are only interested in the right hand tail. We will be using the t-distribution because the population standard deviation is not known.
First we write our hypotheses:
Now we need the appropriate formula for a t-test. We will be using standard error because we are working with the standard deviation of a sampling distribution.
Now we fill in the values from our problem
Now we must look up the t-critical value, or use technology to find the p-value.
We must find the t-critcal value by finding
Because our test statistic 6.32 is more extreme than our critical value, we reject our null hypothesis and conclude that the pretzels do have a higher mean than 1.0.
If you calculated a p-value using technology, p=0.00006884.
Because , we reject our null hypothesis and conclude that the pretzels do have a higher mean than 1.0 g.
This is a one- tailed t-test. It is one-tailed because the question asks whether the pretzel's mean is actually higher, so we are only interested in the right hand tail. We will be using the t-distribution because the population standard deviation is not known.
First we write our hypotheses:
Now we need the appropriate formula for a t-test. We will be using standard error because we are working with the standard deviation of a sampling distribution.
Now we fill in the values from our problem
Now we must look up the t-critical value, or use technology to find the p-value.
We must find the t-critcal value by finding
Because our test statistic 6.32 is more extreme than our critical value, we reject our null hypothesis and conclude that the pretzels do have a higher mean than 1.0.
If you calculated a p-value using technology, p=0.00006884.
Because
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James goes to UCLA, and he believes that the atheletes of UCLA are better runners, than the country average. He did a bit of a research and found that the national average time for a two-mile run for college atheletes is min with a standard deviation of minute. He then sampled UCLA atheletes and found that their average two-mile time was minutes.
Is James' data statistically significant? Can we confirm that UCLA atheletes are better than average runners? And if so, to which level of certainty: , , ,
James goes to UCLA, and he believes that the atheletes of UCLA are better runners, than the country average. He did a bit of a research and found that the national average time for a two-mile run for college atheletes is
Is James' data statistically significant? Can we confirm that UCLA atheletes are better than average runners? And if so, to which level of certainty:
Tap to reveal answer
Using a Z-test (we have population SD, not sample SD) and a population of , we arrive at a P-value of , which is lower than , but above .
Using a Z-test (we have population SD, not sample SD) and a population of
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We are testing the hypothesis that the average gas consumption per day in Billings, Montana is greater than 7 gallons per day; we want 95% confidence.
We sample 30 drivers. The average is 8.4, and the sample standard deviation is 4.29.
Our null hypothesis is
-
What is the Z-value for a 1-tailed test at 95%?
-
What is the Z-value for our sample mean of 8.4?
-
What is the p-value for our sample mean of 8.4?
-
Do we reject the null hypothesis?
We are testing the hypothesis that the average gas consumption per day in Billings, Montana is greater than 7 gallons per day; we want 95% confidence.
We sample 30 drivers. The average is 8.4, and the sample standard deviation is 4.29.
Our null hypothesis is
-
What is the Z-value for a 1-tailed test at 95%?
-
What is the Z-value for our sample mean of 8.4?
-
What is the p-value for our sample mean of 8.4?
-
Do we reject the null hypothesis?
Tap to reveal answer
From the Z-table: 1.8 corresponds to .9641; p= 1 - .9641 = .036
We reject the null hypothesis since 1.8 > 1.64 (and .036 is less than 95%).
In plain English, we are 95% sure that we will not get a sample mean of 8.4 when the true population mean is 7.
From the Z-table: 1.8 corresponds to .9641; p= 1 - .9641 = .036
We reject the null hypothesis since 1.8 > 1.64 (and .036 is less than 95%).
In plain English, we are 95% sure that we will not get a sample mean of 8.4 when the true population mean is 7.
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and the sample mean is 12.
Select the answer so that both statements indicate a rejection of the null hypothesis at the 95% confidence level.
and the sample mean is 12.
Select the answer so that both statements indicate a rejection of the null hypothesis at the 95% confidence level.
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In order to reject the null hypothesis, the Z-value for the sample must be greater than (i.e. must lie outside of) the Z-value of the confidence level.
By definition, if the Z-value of the sample is greater than the Z-value of the confidence level, then the p-value of the sample must be less than the p-value for the confidence level.
In order to reject the null hypothesis, the Z-value for the sample must be greater than (i.e. must lie outside of) the Z-value of the confidence level.
By definition, if the Z-value of the sample is greater than the Z-value of the confidence level, then the p-value of the sample must be less than the p-value for the confidence level.
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