In this lab we’ll conduct two hypothesis testing (HT). The first case considers a population mean, and the second a population proportion. The cases are similar to the examples discussed in class, and follow the steps described in the testing framework.

To help with your assignment below are examples of null and alternative hypotheses written in mathematical notation. The $ signs are introduced to insert a mathematical representation. The backslash \ is used to refer to greek letters and other mathematical symbols. You can copy and edit from the below examples to suit your answers.

Null Hypothesis: \(H_0: \mu = 8\)

Null Hypothesis: \(H_0: p = 0.5\)

Alternative Hypothesis \(H_a: \mu \ne 8\)

Alternative Hypothesis \(H_a: \mu < 8\)

Alternative Hypothesis \(H_a: p \ne 0.5\)

HT for Population Mean

A recent study found out that college students average about 7 hours of sleep per night. However, researchers at an urban college in a big city are interested in showing that their students sleep less than 7 hours on the average. The researchers conducted a simple random sample of n=100 students on campus. They found out that the students averaged 6.6 hours. The previous studies showed that the population standard deviation (\(\sigma\)) of the nightly sleeps was 1.8 hours.

1. Write the null and alternative hypotheses in mathematical notation (2pts)

\(H_0: \mu = 7\), \(H_a: \mu < 7\)

2. Calculate a test statistics and p-value (6pts) Test Statistic Z = 6.6 - 7 / (1.8/ square root of 100) = -2.22

P-value

pnorm(6.6, 7, .1342, T)
## [1] 0.0014383

3. Does this sample provide a convincing evidence that the average sleep time at this urban college is less than 7 hours at a significance level \(\alpha\) of 0.05? Explain your reasoning (2pts)

0.0014383 < 0.05, rejecting the null hypothesis which means that we support our alternative hypothesis. Yes, the sleep time is less than the significance level of 0.05

## HT for Population Proportion

A study examined the average pay for men and women entering the workforce as doctors for 21 different positions. (a) If each gender was equally paid, then we would expect about half of those positions to have men paid more than women and women would be paid more than men in the other half of positions. (b) Men were, on average, paid more in 19 of those 21 positions. Complete a hypothesis test using your hypotheses from part (a).

4. Write the null and alternative hypotheses in mathematical notation (2pts)

\(H_0: p = 0.5\), \(H_a: p \ne 0.5\)

5. Calculate a test statistics and p-value (6pts)

Test Statistic Z = .40/.109 = 3.67

P-value

pnorm(3.67,0,1)
## [1] 0.9998787

6. Does this sample provide a convincing evidence that men are paid more than women at a significance level \(\alpha\) of 0.05? Explain your reasonning (2pts)

Yes men are paid more than women because we reject the null hypothesis and support our alternative hypothesis

This is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was adapted for OpenIntro by Mine Çetinkaya-Rundel from a lab written by the faculty and TAs of UCLA Statistics.

Adapted by Fady Harfoush of Loyola University Chicago for a specific business statistics course.