## Hypothesis Testing Case Solution

Q3) You are curious about whether the professors and students at your school are of different political persuasions, so you take a sample of 20 professors and 20 students drawn randomly from each population. You find that 10 professors say that they are conservative, and 6 students say that they are conservative.

- Describe how you would test whether this is a statistically significant difference. Make sure to calculate the test statistic you propose.
- Suppose that you calculate p-value as 0.196. What is your conclusion?

The null hypothesis states that the two variables are independent from each other and there is no difference between them. In our case, the null hypothesis will state that there is no such difference between students and professors in terms of conservatism. In this scenario, the most appropriate and the recommended test would be t-test as the two variables are not dependent on each other and the level of difference that are found is based on differences that occurred during sampling. For the calculation of value, the significance level is taken as 0.05

t = (0.5 – 0.3) ÷ {√ (0.25/20) + (0.21/20)}

t = 0.200 ÷ 0.152

t = 1.32

Q4) You contact a random sample of 36 graduates of Western University and learn that their starting salaries averaged $28,000 last year. You then contact a random sample of 40 graduates from Eastern University and find that their average starting salary was $28,800. In each case, the standard deviation of the sample was $1,000.

- What assumptions are necessary for a t-test?
- Calculate the test statistic under the null hypothesis that there is no difference between average salaries received by the graduates of the two schools.

a) In order to apply t-test, it is necessary to have sample that is random and taken from ordinary population but with same variance.

b) The null hypothesis in this case would be that no such difference exists between the graduates of two universities in terms of salaries. However, the alternative hypothesis will be that graduates from eastern university are getting higher starting salary as compared to the graduates of western university. The statistical tools that will be used for the hypothesis testing would be t-test as the sample is not dependent on each other. The chosen significance level would be again 0.05 and the calculation has been shown below.

T = {(28800 - 28000) - 0} ÷ {(√36*1000^{2 }+ 40*1000^{2}) * (1 / 36 + 1 / 40)}

T = 800 / 232.81

T = 3.43

Critical value from the table is found to be 1.66 and the calculated value is 3.4 thus, alternative hypothesis is accepted that is graduates from eastern university are getting higher starting salary as compared to the graduates of western university

Q5) A random sample of students are interviewed to determine if there is an association between class and attitude toward corporations. With the following results, how would you test the hypothesis that there is no difference among students on this attitude? Calculate this test statistic.

Favorable | neutral | unfavorable | |

Freshman | 100 | 50 | 70 |

Sophomores | 80 | 60 | 70 |

Juniors | 50 | 50 | 80 |

Seniors | 40 | 60 | 90 |

unfavorable interviewed | % | Favorable | neutral | Unfavorable | |

Freshman | 220 | 27.5 | 74.25 * 100 | 60.5 * 50 | 85.2 * 70 |

Sophomores | 210 | 26.25 | 70.88 * 80 | 57.75 * 60 | 81.3 * 70 |

Juniors | 180 | 22.5 | 60.7 * 50 | 49.5 * 50 | 69.75 * 80 |

Seniors | 190 | 23.75 | 64.1 * 40 | 52.2 * 60 | 73.6 * 90 |

Sum | 100% | 270 | 220 | 310 |

The appropriate sample test would be k sample test that is based on chi-square as the responses are nominal in nature. Significance level for the test would be 0.05 and the calculations have been shown below.

X^{2} = {(25.7^{2} / 74.2) + (9.12^{2} / 70.8) + (-10.7^{2} / 60.7) + (-24.1^{2} / 64.1) + . . . . + (16.3^{2} / 73.6)}

X^{2} = 33.6

The critical value from the table is found to be 12.5 at the significance level of 0.05 and the calculated value is greater thus, leads to rejection of null hypothesis

Q6) You do a survey of business students and liberal arts school students to find out how many times a week they read a daily newspaper. In each case, you interview 100 students. You find the following:

ξm = 4.5 times per week

Sm = 1.5

ξla = 5.6 times per week

S_{la} = - 2.0

How would you test the hypothesis that there is no difference among students on this attitude? Calculate this test statistic.

The alternative hypothesis in this case would be that the readers are different in terms of readership between liberal arts and business students................................

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