Scholarship Examination Scores. At Western University the...

Scholarship Examination Scores. At Western University the historical mean of scholarship examination scores for freshman applications is 900 . A historical population standard deviation $\sigma=180$ is assumed known. Each year, the assistant dean uses a sample of applications to determine whether the mean examination score for the new freshman applications has changed. a. State the hypotheses. b. What is the $95 \%$ confidence interval estimate of the population mean examination score if a sample of 200 applications provided a sample mean $\bar{x}=935$ ? c. Use the confidence interval to conduct a hypothesis test. Using $\alpha=.05$, what is your conclusion? d. What is the $p$-value?

Scholarship Examination Scores. At Western University the historical mean of scholarship examination scores for freshman applications is 900 . A historical population standard deviation $\sigma=180$ is assumed known. Each year, the assistant dean uses a sample of applications to determine whether the mean examination score for the new freshman applications has changed.
a. State the hypotheses.
b. What is the $95 \%$ confidence interval estimate of the population mean examination score if a sample of 200 applications provided a sample mean $\bar{x}=935$ ?
c. Use the confidence interval to conduct a hypothesis test. Using $\alpha=.05$, what is your conclusion?
d. What is the $p$-value?

Key Concepts

Z-Test

A z-test is a type of hypothesis test that is applied when the population standard deviation is known and the sample size is usually large enough for the Central Limit Theorem to apply. It uses the standard normal distribution to calculate a test statistic, which is then compared to critical values to decide whether to reject the null hypothesis.

p-value

The p-value is the probability of obtaining an effect at least as extreme as the one observed in the sample data, assuming that the null hypothesis is true. It quantifies the strength of the evidence against the null hypothesis; a small p-value (typically less than the chosen significance level, such as 0.05) indicates that the observed data are unlikely under the null hypothesis, supporting its rejection.

Hypothesis Testing

Hypothesis testing is a statistical procedure used to decide whether there is enough evidence in a sample of data to infer that a particular condition holds for the entire population. It involves formulating a null hypothesis (H0), which represents a baseline claim (often that there is no effect or no difference), and an alternative hypothesis (Ha), which represents the research claim or effect that is being tested.

Confidence Interval Estimation

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a certain level of confidence (commonly 95%). This interval provides an estimate of uncertainty around the sample estimate, showing both the magnitude and the precision of the estimate.

Transcript

00:01 In this question, we're told that a university entrance exam has average scores of 900 and a standard deviation of 180.

00:09 So we can start by writing that down.

00:12 Population mean is 900.

00:15 Population standard deviation is 180.

00:18 And we're also told that every year the university samples freshmen to determine if the mean score on the entrance exams has changed.

00:27 So for part a, we're supposed to produce hypotheses to test this.

00:32 So the null hypothesis would be that the mean hasn't changed or that the mean is equal to 900.

00:41 And the alternative is that the mean is not equal to 900.

00:48 And then in part b, we're asked to estimate a 95 % confidence interval.

00:53 And we're given that a sample was taken of 200 with a sample mean.

01:01 Of 935.

01:07 So for a 95 % confidence interval, alpha is equal to 1 minus 0 .95, and therefore alpha is equal to 0 .05.

01:18 And so the confidence interval is like this.

01:21 It's the sample average plus or minus the critical value z sub -alph over 2 times the population standard deviation because we're given that, divided by the square root of the sample size.

01:43 And z sub alpha over 2, you can look that up in the z table, or you may have memorized it for alpha equals 0 .05, but that comes out to 1 .96...

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