Critical Thinking on Guam: Statistics Cheatsheet 5

Statistics Cheatsheet 5

by Lei Bao

Hypothesis Tests Regarding a Parameter

	Accept the Null	Reject the Null
Null hypothesis is true	Correct Conclusion	Type I Error
Null hypothesis is false	Type II Error	Correct Conclusion

For students who first enrolled in two year public institutions in a recent semester, the proportion who earned a bachelor’s degree within six years was 0.399. The president of a certain college believes that the proportion of students who enroll in her institution have a higher completion rate.

Determine the null and alternative hypotheses.
- $H_0$ : p = 0.399
- $H_1$ : p > 0.399
Which of the following is a Type I error?
- The president rejects the hypothesis that the proportion of students who earn a bachelor’s degree within six years is 0.399, when, in fact, the proportion is 0.399.
Which of the following is a Type II error?
- The president fails to reject the hypothesis that the proportion of students who earn a bachelor’s degree within six years is 0.399, when, in fact, the proportion is greater than 0.399.

According to a polling organization, 24% of adults in a large region consider themselves to be liberal. A survey asked 200 respondents to disclose their political philosophy: Conservative, Liberal, Moderate. Treat the results of the survey as a random sample of adults in this region. Do the survey results suggest the proportion is higher than that reported by the polling organiztion? Use and $\alpha = 0.10$ level of significance.

The given null and alternative hypotheses.
- $H_0$ : p = 0.24
- $H_1$ : p > 0.24 ( $\hat{p} = 0.35$ )
Find the test statistic, $z_0$ , for this hypothesis test.
- Test Statistics
- $z_0 = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$
- $z_0 = \frac{0.35 - 0.24}{\sqrt{\frac{0.24(1-0.24)}{200}}} = 3.64$
- P-value = 0
- State the conclusion.
  - Reject $H_0$ . There is sufficient evidence at the $\alpha = 0.10$ level of significance to conclude that the proportion is higher than that reported by the polling organization.

The mean waiting time at the drive-throguh of a fast-food restaurant from the time an order is placed to the time the order is received is 83.7 seconds. A manager devises a new drive-through system that she believes will decrease wait time. As a test, she initiates the new system at her restaurant and measures the wait time for 10 randomly selected orders. The wait times are provided in the table to the right.

Because the sample size is small, the manager must verify that the wait time is normally distributed and the sample does not contain any outliers. The sample correlation coefficient is known to be r = 0.977. Are the conditions for testing the hypothesis satisfied?
- Yes, the conditions are satisfied. The normal probability plot is linear enough, since the correlation coefficient is greater than the critical value.
Is the new system effective? Conduct a hypothesis test using the P-value approach and a level of significance of $\alpha = 0.05$ .
- $H_0$ : $\mu = 83.7$
- $H_1$ : $\mu < 83.7$ ($\bar{x} = 78 $, $s_x = 13.973$ )
- $t_0 = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$
- $t_0 = \frac{78 - 83.7}{\frac{13.973}{\sqrt{10}}} = -1.29$
- P-value
- 0.9015
Use the $\alpha = 0.05$ level of significance. What can be concluded from the hypothesis test?
- The P-value is greater than the level of significance so there is not sufficient evidence to conclude the new system is effective.

An educator wants to determine whether a new curriculum significantly improve standardized test scores for third grade students. She randomly divides 80 third-graders into two groups. Group 1 is taught using the tradidtional curriculum, while group 2 is taught using the new curriculum. At the end of the school year, both groups are given the standardized test and the mean scores are compared. Determine whether the sampling is dependent or indepdendent. Indicate whether the response variable is qualitative or quantitative.

This sampling is independent because the individuals selected for one sample do not dictate which individuals are to be in a second sample.
The variable is quantitative because it is a numerical measure.

In 1945, an organization surveyed 1100 adults and asked, “Are you total abstainer from, or do you on occasion consume, alcoholic beverages?” Of the 1100 adults surveyed, 396 indicated that they were total abstainers. In a recent survey, the same question was asked of 1100 adults and 319 indicated that they were total abstainers.

Determine the sample proportion for each sample.
- The proportions of the adults who took the 1945 survey and the recent survey who were total abstainers are 0.36 and 0.29, respectively.
Has the proportion of adults who totally abstain from alcohol changed? Use the $\alpha = 0.01$ level of significance.
First verify the model requirements.
- $n_1 \hat{p}_1 (1 - \hat{p}_1) \ge 10$ and $n_2 \hat{p}_2 (1 - \hat{p}_2) \ge 10$
Identify the null and alternative hypotheses for this test. Let $p_1$ represent the population proportion of 1945 adults who were total abstainers and $p_2$ represent the population proportion of recent adults who were total abstainers.
- $H_0$ : $p_1 = p_2$
- $H_1$ : $p_1 \ne p_2$
- $p = 0.325$
- $p_1 = 0.36$
- $p_2 = 0.29$
Find the test statistic for this hypothesis test
$z_0 = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p} (1- \hat{p})} \sqrt{1/n_1 + 1/n_2}}$
$z_0 = \frac{0.36 - 0.29}{\sqrt{0.325 (1- 0.325)} \sqrt{1/1100 + 1/1100}} = 3.50$
Determine the P-value for this hypothesis test.
- P-value = 0
Interpret the P-value
- If the population proportions are equal, one would expect a sample difference proportion greater than the absolute value of the one observed in about 0 out of 100 repetitions of this experiment.
State the conclusion for this hypothesis test.
- Reject $H_0$ . There is sufficient evidence at the $\alpha = 0.01$ level of significance to suggest the proportion of adults who totally abstain from alcohol has changed.