# 5.2.1 - Examples

5.2.1 - Examples## Example: Rent

**Research question**: Is the average monthly rent of a one-bedroom apartment in State College, Pennsylvania less than \$900?

In this question we are comparing the mean of all State College one-bedroom apartments (i.e. \(\mu\)) to the value of \$900. This is a single sample mean test. We want to know if the population mean is less than \$900, so this is a left-tailed test. Our hypotheses are:

- \(H_0:\mu=\$900\)
- \(H_a: \mu < \$900\)

## Example: IQ Scores

**Research question**: Is the average IQ score of all World Campus STAT 200 students higher than the national average of 100?

In this question we are comparing the mean of all World Campus STAT 200 students (i.e. \(\mu\)) to the given value of 100. This is a single sample mean test. We want to know if the population mean is greater than 100, so this is a right-tailed test. Our hypotheses are:

- \(H_0:\mu = 100\)
- \(H_a: \mu > 100\)

## Example: Weight Loss

**Research question: **Do participants lose weight following a weight-loss intervention?

Data were collected from one group of participants before and after a weight-loss intervention. Data were paired by participant. Assuming that \(x_1\) is an individual's weight before the intervention and \(x_2\) is their weight at the end of the study, if they lost weight then \(x_1-x_2\) would be a positive number (i.e., greater than 0). Thus, this is a right-tailed test. Because we are testing their mean difference, the parameter that we should write in our hypotheses is \(\mu_d\) where \(\mu_d\) is the mean weight change (before-after) in the population.

Our hypotheses are:

- \(H_0: \mu_d=0\)
- \(H_a:\mu_d > 0 \)

## Example: Gender of College of Science Students

**Research question**: Is the percent of students enrolled in Penn State's College of Science who identify as women different from 50%?

In this question we are comparing the proportion of all Penn State College of Science students (i.e. \(p\)) to the given value of 0.5. This is a single sample proportion test. We want to know if the population proportion is different from 0.5, so this is a two-tailed test. Our hypotheses are:

- \(H_0:p=0.5\)
- \(H_a: p ≠ 0.5\)

## Example: Dog Ownership

**Research question**: Do the majority of all World Campus STAT 200 students own a dog?

If the majority of all students own a dog, then more than 50% own a dog. In this question we are comparing the population proportion for all World Campus STAT 200 students (i.e. \(p\)) to the value of 0.5. This is a single sampling proportion test. We want to know if the proportion is greater than 0.5, so this is a right-tailed test. Our hypotheses are:

- \(H_0:p=0.5\)
- \(H_a: p > 0.5\)

## Example: Weights of Boys and Girls

**Research question**: In preschool, are the weights of boys and girls different?

We are comparing the weights of two independent groups: boys and girls. Weight is a quantitative variable so the parameter we are testing is \(\mu\). Our research question does not hypothesize which group has the larger weight, so this is a two-tailed test. Our hypotheses are:

- \(H_0: \mu_b = \mu_g\)
- \(H_a: \mu_b \ne \mu_g\)

*Note: This is equivalent to \(H_0: \mu_b - \mu_g = 0\) and \(H_a: \mu_b - \mu_g \ne 0\). *

## Example: Smoking by Gender

**Research question**: Is the proportion of men who smoke cigarettes different from the proportion of women who smoke cigarettes in the United States?

In this question we are comparing two independent groups: men and women. The response variable, smoking, is categorical therefore we are comparing proportions. Our research question does not suggest which group smokes more, so we have a two-tailed test. Our hypotheses are:

- \(H_0: p_1=p_2\)
- \(H_a: p_1 \ne p_2\)

*Note: This is equivalent to \(H_0: p_1 - p_2 =0\) and \(H_a: p_1 - p_2 \ne 0\)*

## Example: Predicting SAT-Math using IQ

**Research question**: Can IQ scores be used to predict SAT-Math scores in the population of all American high school seniors?

SAT-Math and IQ scores are both quantitative variables. Our research question is about prediction, so we are going to use simple linear regression. The parameter we are testing is \(\beta\). Our research question does not state whether we expect the slope to be positive or negative, therefore this is a two-tailed test. Our hypotheses are:

- \(H_0: \beta = 0\)
- \(H_a: \beta \ne 0\)

## Example: Relation Between Height and Weight

**Research question**: Is there a positive relationship between height and weight in the population of all American adults age 25 and older?

The relationship between two quantitative variables is measured using correlation (Pearson's r). The parameter we are testing is \(\rho\). A positive relationship would be indicated by a positive correlation coefficient, therefore this is a right-tailed test. Our hypotheses are:

- \(H_0: \rho = 0\)
- \(H_a: \rho > 0\)