Question

A large school in Michigan is concerned that too many of the Seniors have contracted the Senioritis-20 virus. THe administrators have decided that if more the 30% of the Seniors have the virus, they will cancel Spring Break. To make this decision, they will select a random sample of 50 Seniors and have them tested.

1. State the appropriate hypotheses for performing a significance test with an alpha level of 0.05.

2. (a) Describe a Type I error in this setting; (b) Describe a Type II error in this setting; (c) From the administrator's perspective, which error has more serious consequences?

3-4. Compare how the graph of the sampling distribution changes when we assume Ho is true versus when the Ha is true (ie. if the truepercentage of Senioritis is 50%)

5. In this setting, the power of the test is 0.91. Interpret the power of the test.

6. Use the applet provided below to analyze the data in this problem. Describe what you discovered.

7. The administration wants to increase the power of the test. How could they adjust each of the following factors to increase the power? (a) Sample Size (b) alpha level (c) Alternative p

Answer 1

As we are concerned about the proportion of population contracting this seniority virus.

lets denote it by p1

1) Null hypothesis: p1=30 proportion of seniors contracting virus is 30.

Alternative : p1>30 (proportion of seniors contracting is greater than 30)

2) Type 1 error- rejecting Null hypothesis when it's true.

In this setting administration will commit type 1 error if they reject null hypothesis when its true i.e if they conclude that more than 30% of seniors have contracted the virus when its actually false and they cancel spring break.

Type 2 error- Accept null hypothesis when its actually false.

from administrations perspective type 2 is committed when they conclude p1=30 i.e 30% have contracted the virus when its actually greater than 30%.

now which is more serious really depends on the situation generally type -1 is deemed to be more serious.

if 0administration commits type 1 error then spring break will be canceled for no reason.

3-4) when H0 is true mean will be 0.30 graph will centred around this. under h1 when p1>0.30 it will tend to shift right centering around 0.50 as it is the true mean.

5)Power of test is probability of rejecting null hypothesis when its actually false.So a test how correctly reject a null hypothesis when it is false. It's a measure of goodness of test. in this setting power is 0.91 so it mean 91% of the time test will correctly reject the null hypothesis. i.e 91% of the time administration will correctly conclude that seniority virus is greater than 30% which will lead to cancellation of spring break.

6) applet is not there so this can't be answered.

7)

a)All things being equal greater the sample size greater the power of test so they can increase sample size.

b)The lower the significance level, the lower the power of the test. If you reduce the significance level (e.g., from 0.05 to 0.01), the region of acceptance gets bigger. As a result, you are less likely to reject the null hypothesis. This means you are less likely to reject the null hypothesis when it is false, so you are more likely to make a Type II error. In short, the power of the test is reduced when you reduce the significance level; and vice versa.

c)The greater the difference between the "true" value of a parameter and the value specified in the null hypothesis, the greater the power of the test. That is, the greater the effect size, the greater the power of the test.

that means if p1=0.20 instead of 0.30 power of test will increase.