It is advertised that the average braking distance for a small car traveling at 65 miles per hour is 120 feet. A transportation researcher wants to determine if this statement is false, meaning it differs from the advertised distance. She randomly test drives 36 small cars at 65 miles per hour and records the braking distance. The sample average braking distance is 126 feet. Assume that the population standard deviation is 22 feet.

1. State the null and alternative hypotheses.
- Null Hypothesis ($H_0$): The average braking distance is 120 feet.
- Alternative Hypothesis ($H_a$): The average braking distance is not 120 feet.

2. Using the critical value approach, test the hypotheses at the 5% level of significance. What is the calculated test statistic for this problem?

Multiple Choice:
A. -1.64
B. +1.64
C. -2.20
D. +2.20

Null Hypothesis [tex](H_0)[/tex]: The average braking distance [tex](\mu)[/tex] is equal to 120 feet. [tex]H_0: \mu = 120[/tex]
Alternative Hypothesis [tex](H_a)[/tex]: The average braking distance [tex](\mu)[/tex] is different from 120 feet. [tex]H_a: \mu \neq 120[/tex]

This is a two-tailed test because we are checking if the braking distance is different (it could be either more or less).

Step 2: Identify Key Information

Sample size [tex](n)[/tex]: 36
Sample mean [tex](\bar{x})[/tex]: 126 feet
Population standard deviation [tex](\sigma)[/tex]: 22 feet
Significance level [tex](\alpha)[/tex]: 0.05

Step 3: Calculate the Test Statistic

The formula for the test statistic [tex]z[/tex] is:

[tex]z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}[/tex]

Where:

[tex]\bar{x}[/tex] = Sample mean = 126
[tex]\mu_0[/tex] = Hypothesized population mean = 120
[tex]\sigma[/tex] = Population standard deviation = 22
[tex]n[/tex] = Sample size = 36

Substitute the values into the formula:

[tex]z = \frac{126 - 120}{\frac{22}{\sqrt{36}}}[/tex]

[tex]z = \frac{6}{\frac{22}{6}}[/tex]

[tex]z = \frac{6}{3.6667}[/tex]

[tex]z \approx 1.64[/tex]

Step 4: Determine the Critical Value(s)

For a two-tailed test at a significance level of 0.05, the critical z-values are approximately [tex]-1.96[/tex] and [tex]1.96[/tex]. This means we will reject the null hypothesis if the calculated test statistic is less than [tex]-1.96[/tex] or greater than [tex]1.96[/tex].

Step 5: Make the Decision

Since the calculated test statistic [tex]z \approx 1.64[/tex] is not greater than [tex]1.96[/tex] or less than [tex]-1.96[/tex], we fail to reject the null hypothesis.

This means there is not enough statistical evidence to say that the true average braking distance is different from 120 feet at the 5% significance level.

Thus, the correct answer is +1.64.