For Exercises 5 through 20, perform each of the following...

For Exercises 5 through $20,$ perform each of the following steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Undergraduate Enrollment It has been found that $85.6 \%$ of all enrolled college and university students in the United States are undergraduates. A random sample of 500 enrolled college students. A random sample of 500 enrolled college students in a particular state revealed that 420 of them were undergraduates. Is there sufficient evidence to conclude that the proportion differs from the national percentage? Use $\alpha=0.05 .$

For Exercises 5 through $20,$ perform each of the following steps.
a. State the hypotheses and identify the claim.
b. Find the critical value(s).
c. Compute the test value.
d. Make the decision.
e. Summarize the results.
Use the traditional method of hypothesis testing unless otherwise specified.
Undergraduate Enrollment It has been found that $85.6 \%$ of all enrolled college and university students in the United States are undergraduates. A random sample of 500 enrolled college students. A random sample of 500 enrolled college students in a particular
state revealed that 420 of them were undergraduates. Is there sufficient evidence to conclude that the proportion differs from the national percentage? Use $\alpha=0.05 .$

Key Concepts

Hypothesis Testing Framework

This concept involves setting up competing hypotheses to answer a research question. It requires the formulation of a null hypothesis, which generally represents no effect or a status quo claim, and an alternative hypothesis that represents the claim being tested. The framework provides a structured process to decide whether to reject the null hypothesis in favor of the alternative based on sample data and a pre-determined significance level.

Population Proportion Test

This key concept relates to testing claims about a single population proportion. It involves using a sample to estimate the proportion of successes (or a particular characteristic) and comparing it to a known or hypothesized proportion in the overall population. The test uses the normal approximation to the binomial distribution when the sample size is large enough, which allows for the use of z-tests.

Critical Value Approach

This approach involves determining threshold values, based on the chosen significance level (alpha), that define the boundaries for the rejection region of the test statistic. If the computed test statistic falls beyond these critical values, the null hypothesis is rejected. This method is central to the traditional hypothesis testing procedure.

Test Statistic Calculation

This concept involves computing a standardized statistic (such as a z-score) that measures how far the observed sample proportion deviates from the hypothesized proportion, in terms of the standard error. The computed test statistic is then compared to the critical value(s) to determine if the observed discrepancy is statistically significant.

Decision Rule

The decision rule is the criterion that specifies under what conditions the null hypothesis will be rejected. It is based on the relationship between the test statistic and the critical values derived from the significance level. If the test statistic falls within the rejection region, the null hypothesis is rejected, which in turn may indicate support for the alternative hypothesis.

Interpretation and Summary of Results

This key concept involves drawing a conclusion from the hypothesis test by relating the statistical decision back to the real-world context of the problem. It requires a clear summary of the findings, including whether there is sufficient evidence to support the claim made by the alternative hypothesis, and discussing the implications of the result in the context of the original research question.

Transcript

00:04 This problem, we are told that 85 .6 % of all students enrolled in university are undergrads.

00:18 We want to check this, so we have a random sample of 500 college students, and we find that 420 of them are undergrads.

00:40 The claim is that the true proportion of university students that are undergrads is 80.

01:09 5 .6 % they want to see.

01:18 Are we, our hypothesis is that it is different from 0 .856.

01:46 We now need to find our critical value.

01:51 We are given an alpha level of 0 .05, and we know this is a two -tailed test, test since we are looking at if it's different than you're looking at it's not equal to.

02:07 So that means it's a two -tailed.

02:18 So we have an area on either side.

02:23 Each area is equal to 025.

02:26 So we find a critical value of plus or minus 1 .96.

02:34 Next we need to find our test values.

03:00 We'll start by finding our sample proportion, p hat, just given as x over n, our our result over our sample size, and our result is 420 over 500.

03:16 That is equal to a proportion of 0 .84.

03:30 We now need to find our z value.

03:37 C is given as p -hat minus p over the square root.

03:46 Of p times 1 minus p, also known as q over n.

03:52 So we have 0 .84 minus 0 .856 over 0 .856 times 1 minus 0 .856 all over 500.

04:16 We get an answer of negative 1 .02.

04:39 So now we need to make our decision of whether or not we can reject or if we fail to reject our claim or the null hypothesis...

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