Hypothesis Tests for a Single Population Proportion

1. You are conducting a study to see if the proportion of men over the age of 50 who regularly have their prostate examined is significantly less than 0.33. A random sample of 761 men over the age of 50 found that 205 have their prostate regularly examined. Do the sample data provide convincing evidence to support the claim? Test the relevant hypotheses using a 5% level of significance. Give answer to at least 4 decimal places. a. What are the correct hypotheses? (Select the correct symbols and use decimal values not percentages.) H0: H1: Based on the hypotheses, find the following: b. Test Statistic = c. Critical-value= d. Shade the sampling distribution curve with the correct critical value(s) and shade the critical regions. The arrows can only be dragged to z-scores that are accurate to 1 place after the decimal point (these values correspond to the tick marks on the horizontal axis). Select from the drop down menu to shade to the left, to the right, between or left and right of the z-score(s). Shade: . Click and drag the arrows to adjust the values. -1.4 e. The correct decision is to . f. The correct summary would be: that the proportion of men over the age of 50 who regularly have their prostate examined is significantly less than 0.33. 2. Test the claim that the proportion of people who own cats is smaller than 20% at the 0.10 significance level. The null and alternative hypothesis would be: The test is: two-tailed right-tailed left-tailed   Based on a sample of 500 people, 16% owned cats The test statistic is: (to 2 decimals) The p-value is: (to 2 decimals) Based on this we: • Reject the null hypothesis • Fail to reject the null hypothesis 3. A well-known brokerage firm executive claimed that 20% of investors are currently confident of meeting their investment goals. An XYZ Investor Optimism Survey, conducted over a two week period, found that in a sample of 700 people, 18% of them said they are confident of meeting their goals. Test the claim that the proportion of people who are confident is smaller than 20% at the 0.05 significance level. The null and alternative hypothesis would be: The test is: two-tailed left-tailed right-tailed The test statistic is: (to 3 decimals) The p-value is: (to 4 decimals) Based on this we: • Fail to reject the null hypothesis • Reject the null hypothesis 4. Test the claim that the proportion of people who own cats is larger than 20% at the 0.025 significance level. The null and alternative hypothesis would be: The test is: • left-tailed • right-tailed • two-tailed Based on a sample of 800 people, 23% owned cats The test statistic is: (Round to 2 decimals) The p-value is: (Round to 2 decimals) Based on this we: • Reject the null hypothesis • Do not reject the null hypothesis 5. Only 20% of registered voters voted in the last election. Will voter participation decline for the upcoming election? Of the 388 randomly selected registered voters surveyed, 58 of them will vote in the upcoming election. What can be concluded at the = 0.01 level of significance? a. For this study, we should use b. The null and alternative hypotheses would be: (please enter a decimal) (Please enter a decimal) c. The test statistic • = (please show your answer to 3 decimal places.) • The p-value = (Please show your answer to 4 decimal places.) • The p-value is Based on this, we should • the null hypothesis. • Thus, the final conclusion is that ... • The data suggest the population proportion is not significantly lower than 20% at • = 0.01, so there is statistically significant evidence to conclude that the percentage of registered voters who will vote in the upcoming election will be equal to 20%. • The data suggest the population proportion is not significantly lower than 20% at • = 0.01, so there is statistically insignificant evidence to conclude that the percentage of registered voters who will vote in the upcoming election will be lower than 20%. • The data suggest the populaton proportion is significantly lower than 20% at = 0.01, so there is statistically significant evidence to conclude that the the percentage of all registered voters who will vote in the upcoming election will be lower than 20%. 6. Test the claim that the proportion of people who own cats is larger than 40% at the 0.1 significance level.

H0: p = 0.33
H1: p < 0.33

z = (205 - 761 * 0.33) / √(761 * 0.33 * 0.67) = -1.40

zα = -1.645

H0: p = 0.2
H1: p < 0.2

z = (16 - 500 * 0.2) / √(500 * 0.2 * 0.8) = -2.58

p = 0.005

H0: p = 0.2
H1: p < 0.2

z = (18 - 700 * 0.2) / √(700 * 0.2 * 0.8) = -1.73

p = 0.082

H0: p = 0.2
H1: p > 0.2

z = (23 - 800 * 0.2) / √(800 * 0.2 * 0.8) = 2.31

p = 0.019