MTH 128 STATICS

Mth 128

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

1. 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. The alternative hypothesis would be:

The test is:

right-tailed

two-tailed

left-tailed

Based on a sample of 600 people, 44% 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

The conclusion is:

· At the 0.1 alpha level, I was able to show that the proportion of people who own cats is significantly less than 40 percent.

· At the 0.1 alpha level, I was able to show that the proportion of people who own cats is equal to 40 percent.

· At the 0.1 alpha level, I was able to show that the proportion of people who own cats is significantly greater than 40 percent.

· At the 0.1 alpha level, I was unable to show that the proportion of people who own cats is significantly different than 40 percent.

· At the 0.1 alpha level, I was unable to show that the proportion of people who own cats is significantly greater than 40 percent.

· At the 0.1 alpha level, I was unable to show that proportion of people who own cats is significantly less than 40 percent.

7. 16% of all college students volunteer their time. Is the percentage of college students who are volunteers smaller for students receiving financial aid? Of the 352 randomly selected students who receive financial aid, 49 of them volunteered their time. What can be concluded at the 

= 0.01 level of significance?

a. For this study, we should use

1. 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 populaton proportion is significantly lower than 16% at

 = 0.01, so there is sufficient evidence to conclude that the percentage of financial aid recipients who volunteer is lower than 16%.

 The data suggest the population proportion is not significantly lower than 16% at

 = 0.01, so there is insufficient evidence to conclude that the percentage of financial aid recipients who volunteer is lower than 16%.

 The data suggest the population proportion is not significantly lower than 16% at

1.

0. = 0.01, so there is sufficient evidence to conclude that the percentage of financial aid recipients who volunteer is equal to 16%.

8. In a certain school district, it was observed that 27% of the students in the element schools were classified as only children (no siblings).  However, in the special program for talented and gifted children, 74 out of 226 students are only children.  The school district administrators want to know if the proportion of only children in the special program is significantly different from the proportion for the school district. Test at the level of significance. What is the hypothesized population proportion for this test?

(Report answer as a decimal accurate to 2 decimal places.  Do not report using the percent symbol.) Based on the statement of this problem, how many tails would this hypothesis test have?

· one-tailed test

· two-tailed test

Choose the correct pair of hypotheses for this situation:

Using the normal approximation for the binomial distribution (without the continuity correction), was is the test statistic for this sample based on the sample proportion? (Report answer as a decimal accurate to 3 decimal places.) You are now ready to calculate the P-value for this sample. P-value = (Report answer as a decimal accurate to 4 decimal places.) This P-value (and test statistic) leads to a decision to...

· reject the null

· accept the null

· fail to reject the null

· reject the alternative

As such, the final conclusion is that...

· There is sufficient evidence to warrant rejection of the assertion that there is a different proportion of only children in the G&T program.

· There is not sufficient evidence to warrant rejection of the assertion that there is a different proportion of only children in the G&T program.

· The sample data support the assertion that there is a different proportion of only children in the G&T program.

· There is not sufficient sample evidence to support the assertion that there is a different proportion of only children in the G&T program.

9. In a certain school district, it was observed that 32% of the students in the element schools were classified as only children (no siblings).  However, in the special program for talented and gifted children, 84 out of 214 students are only children.  The school district administrators want to know if the proportion of only children in the special program is significantly different from the proportion for the school district. Test at the level of significance. What is the hypothesized population proportion for this test?

(Report answer as a decimal accurate to 2 decimal places.  Do not report using the percent symbol.) Based on the statement of this problem, how many tails would this hypothesis test have?

· one-tailed test

· two-tailed test

11 A medical researcher says that less than 25% of U.S. adults are smokers. In a simple random sample of 200 U.S. adults, 20% were smokers. Can we conclude that less than 25% of U.S. adults are smokers? Use 1% to decide. Round to the fourth as needed What's the minimum population size required? How many successes were there? : : Test Statistic: P-value: Did something significant happen? Select the Decision Rule: There enough evidence to conclude

12 The US Department of Energy reported that 48% of homes were heated by natural gas. A random sample of 303 homes in Oregon found that 140 were heated by natural gas. Test the claim that proportion of homes in Oregon that were heated by natural gas is different than what was reported. Use a 1% significance level. 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, compute the following: b. Test Statistic = c. p-value =

d. Based on the above we choose to

e. The correct summary would be: that the proportion of homes in Oregon that were heated by natural gas is different than what the DOE reported value of 48%.

13. The US Department of Energy reported that 48% of homes were heated by natural gas. A random sample of 303 homes in Oregon found that 140 were heated by natural gas. Test the claim that proportion of homes in Oregon that were heated by natural gas is different than what was reported. Use a 1% significance level. 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, compute the following: b. Test Statistic = c. p-value =

d. Based on the above we choose to

e. The correct summary would be: that the proportion of homes in Oregon that were heated by natural gas is different than what the DOE reported value of 48%.

14. You are conducting a study to see if the proportion of voters who prefer the Democratic candidate is significantly different from 51% at a level of significance of

= 0.05. According to your sample, 40 out of 76 potential voters prefer the Democratic candidate.

a. For this study, we should use

The null and alternative hypotheses would be:      Ho: (please enter a decimal)     H1:

1. (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 different from 51% at

 = 0.05, so there is sufficient evidence to conclude that the proportion of voters who prefer the Democratic candidate is equal to 51%.

 The data suggest the population proportion is not significantly different from 51% at

 = 0.05, so there is not sufficient evidence to conclude that the proportion of voters who prefer the Democratic candidate is different from 51%.

 The data suggest the populaton proportion is significantly different from 51% at

1.

0. = 0.05, so there is sufficient evidence to conclude that the proportion of voters who prefer the Democratic candidate is different from 51%

1. Interpret the p-value in the context of the study.

1. There is a 77.6% chance that the percent of all voters who prefer the Democratic candidate differs from 51%.

1. If the population proportion of voters who prefer the Democratic candidate is 51% and if another 76 voters are surveyed then there would be a 77.6% chance that either more than 53% of the 76 voters surveyed prefer the Democratic candidate or fewer than 49% of the 76 voters surveyed prefer the Democratic candidate.

1. There is a 77.6% chance of a Type I error.

1. If the sample proportion of voters who prefer the Democratic candidate is 53% and if another 76 voters are surveyed then there would be a 77.6% chance that we would conclude either fewer than 51% of all voters prefer the Democratic candidate or more than 51% of all voters prefer the Democratic candidate.

1. Interpret the level of significance in the context of the study.

2. If the population proportion of voters who prefer the Democratic candidate is 51% and if another 76 voters are surveyed then there would be a 5% chance that we would end up falsely concluding that the proportion of voters who prefer the Democratic candidate is different from 51%

2. There is a 5% chance that the earth is flat and we never actually sent a man to the moon.

2. If the proportion of voters who prefer the Democratic candidate is different from 51% and if another 76 voters are surveyed then there would be a 5% chance that we would end up falsely concluding that the proportion of voters who prefer the Democratic candidate is equal to 51%.

2. There is a 5% chance that the proportion of voters who prefer the Democratic candidate is different from 51%.

15. 9% of all Americans suffer from sleep apnea. A researcher suspects that a higher percentage of those who live in the inner city have sleep apnea. Of the 388 people from the inner city surveyed, 50 of them suffered from sleep apnea. What can be concluded at the level of significance of

= 0.01?

a. For this study, we should use

The null and alternative hypotheses would be:      Ho: (please enter a decimal)     H1:

1. (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 populaton proportion is significantly larger than 9% at

 = 0.01, so there is sufficient evidence to conclude that the population proportion of inner city residents who have sleep apnea is larger than 9%

 The data suggest the population proportion is not significantly larger than 9% at

 = 0.01, so there is not sufficient evidence to conclude that the population proportion of inner city residents who have sleep apnea is larger than 9%.

 The data suggest the population proportion is not significantly larger than 9% at

1.

0. = 0.01, so there is sufficient evidence to conclude that the population proportion of inner city residents who have sleep apnea is equal to 9%.

1. Interpret the p-value in the context of the study.

1. There is a 0.37% chance of a Type I error.

1. If the population proportion of inner city residents who have sleep apnea is 9% and if another 388 inner city residents are surveyed then there would be a 0.37% chance that more than 13% of the 388 inner city residents surveyed have sleep apnea.

1. There is a 0.37% chance that more than 9% of all inner city residents have sleep apnea.

1. If the sample proportion of inner city residents who have sleep apnea is 13% and if another 388 inner city residents are surveyed then there would be a 0.37% chance of concluding that more than 9% of all inner city residents have sleep apnea.

1. Interpret the level of significance in the context of the study.

2. If the population proportion of inner city residents who have sleep apnea is 9% and if another 388 inner city residents are surveyed then there would be a 1% chance that we would end up falsely concluding that the proportion of all inner city residents who have sleep apnea is larger than 9%.

2. There is a 1% chance that aliens have secretly taken over the earth and have cleverly disguised themselves as the presidents of each of the countries on earth.

2. If the population proportion of inner city residents who have sleep apnea is larger than 9% and if another 388 inner city residents are surveyed then there would be a 1% chance that we would end up falsely concluding that the proportion of all inner city residents who have sleep apnea is equal to 9%.

2. There is a 1% chance that the proportion of all inner city residents who have sleep apnea is larger than 9%.