Carrying Out Test for Population Mean - AP Statistics

What does $n$ represent in hypothesis testing?

Sample size. Number of observations in the sample.
What is the role of the alternative hypothesis ( $H_a$ )?

Proposes the opposite of the null hypothesis. Represents researcher's theory to be supported.
Which option is a requirement for the Central Limit Theorem?

Sample size $n$ is sufficiently large. Enables normal approximation for sampling distribution.
What is the critical value approach?

Compare test statistic to critical value to decide on $H_0$ . Uses predetermined threshold to make statistical decision.
Choose the condition that must be met for a t-test.

Random sample or normal distribution. Ensures valid statistical inference assumptions.
Identify the degrees of freedom for a t-test.

$n - 1$ . Sample size minus one for t-distribution.
What does a Type I error mean in hypothesis testing?

Rejecting $H_0$ when it is true. False positive error in statistical testing.
What is the relationship between confidence level and confidence interval width?

Higher confidence level results in wider interval. Trade-off between confidence and precision.
What is power in the context of hypothesis testing?

Probability of rejecting $H_0$ when it is false. Ability to detect true effects when they exist.
Which option is affected by the choice of $\alpha$ ?

Significance level and Type I error rate. Alpha determines probability thresholds for decisions.
What is the sample mean ( $\bar{x}$ ) used in hypothesis testing?

Average of sample data points. Central measure of sample location.
What is the consequence of a Type II error?

Failing to detect a real effect. Missing a real effect that exists.
What is the impact of a small p-value?

Evidence against $H_0$ . Low probability suggests null hypothesis is false.
What is the sample standard deviation ( $s$ )?

Measure of variation among sample data points. Estimates population standard deviation from sample.
What is the purpose of a hypothesis test for a population mean?

To determine if sample mean differs from population mean. Tests whether sample provides evidence of population difference.
What is the decision rule for rejecting $H_0$ using p-value?

Reject $H_0$ if p-value $< \alpha$ . Standard decision criterion for hypothesis testing.
Which option is a benefit of a larger sample size?

More precise estimate of population mean. Reduced sampling error improves estimation accuracy.
What does a Type II error mean in hypothesis testing?

Failing to reject $H_0$ when it is false. False negative error in statistical testing.
What does the term 'statistically significant' imply?

Results unlikely due to chance at given $\alpha$ . Unlikely to occur by random chance alone.
Calculate the test statistic: $\bar{x} = 98$ , $\mu = 100$ , $s = 5$ , $n = 25$ .

$t = -2$ . Using $t = \frac{98-100}{5/\sqrt{25}} = \frac{-2}{1} = -2$ .
Which option is a correct interpretation of a 95% confidence interval?

95% of such intervals contain the population mean. Long-run interpretation of confidence level.
Identify the condition of normality for a t-test.

Population is normally distributed or $n > 30$ . Ensures valid use of t-distribution for inference.
What is the consequence of a Type I error?

Falsely claiming a significant effect. Incorrectly rejecting a true null hypothesis.
Which option defines the significance level ( $\alpha$ )?

Probability of rejecting $H_0$ when it is true. Controls Type I error rate in hypothesis testing.
What happens to the test statistic if the sample mean increases?

Test statistic increases. Larger difference from null increases test statistic.
What is the critical region in hypothesis testing?

Range of values for which $H_0$ is rejected. Values where null hypothesis gets rejected.
What is the effect of increasing sample size on the confidence interval?

Narrower confidence interval. Larger sample reduces standard error.
What happens to the power of a test if $\alpha$ is increased?

Power increases. Higher alpha makes rejection easier.

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