MATHEMATICS

The screening process for detecting a rare disease is not perfect. Researchers have developed a blood test that is considered fairly reliable. It gives a positive reaction in 98% of the people who have that disease. However, it erroneously gives a positive reaction in 3% of the people who do not have the disease. Answer the following questions using the null hypothesis as “the individual does not have the disease.”
1.

a. What is the probability of Type I error? (Round your answer to 2 decimal places.)

Probability

b. What is the probability of Type II error? (Round your answer to 2 decimal places.)

Probability

2.
Consider the following hypotheses:

H0: μ ≤ 12.6
HA: μ > 12.6

A sample of 25 observations yields a sample mean of 13.4. Assume that the sample is drawn from a normal population with a known population standard deviation of 3.2. Use Table 1.

a. Calculate the p-value. (Round “z” value to 2 decimal places and final answer to 4 decimal places.)

p-value

b. What is the conclusion if α = 0.10?

Reject H0 since the p-value is greater than α.

Reject H0 since the p-value is smaller than α.

Do not reject H0 since the p-value is greater than α.

Do not reject H0 since the p-value is smaller than α.

c. Calculate the p-value if the above sample mean was based on a sample of 100 observations. (Round “z” value to 2 decimal places and final answer to 4 decimal places.)

p-value

d. What is the conclusion if α = 0.10?

Reject H0 since the p-value is greater than α.

Reject H0 since the p-value is smaller than α.

Do not reject H0 since the p-value is greater than α.

Do not reject H0 since the p-value is smaller than α.

3.
Consider the following hypotheses:

H0: μ ≥ 150
HA: μ < 150

A sample of 80 observations results in a sample mean of 144. The population standard deviation is known to be 28. Use Table 1.

a. What is the critical value for the test with α = 0.01 and with α = 0.05? (Negative values should be indicated by a minus sign. Round your answers to 2 decimal places.)

Critical Value
α = 0.01

α = 0.05

________________________________________

b-1. Calculate the value of the test statistic. (Negative value should be indicated by a minus sign. Round intermediate calculations to 4 decimal places. Round your answer to 2 decimal places.)

Test statistic

b-2. Does the above sample evidence enable us to reject the null hypothesis at α = 0.01?

Yes since the value of the test statistic is not less than the negative critical value.

Yes since the value of the test statistic is less than the negative critical value.

No since the value of the test statistic is not less than the negative critical value.

No since the value of the test statistic is less than the negative critical value.

c. Does the above sample evidence enable us to reject the null hypothesis at α = 0.05?

Yes since the value of the test statistic is not less than the negative critical value.

Yes since the value of the test statistic is less than the negative critical value.

No since the value of the test statistic is not less than the negative critical value.

No since the value of the test statistic is less than the negative critical value.

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