Questions: Daily Driving The average number of miles a person drives per day is 24. A researcher wishes to see if people over age 60 drive less than 24 miles per day. She selects a random sample of 35 drivers over the age of 60 and finds that the mean number of miles driven is 23.2. The population standard deviation is 3 miles. At α=0.01, is there sufficient evidence that those drivers over 60 years old drive less than 24 miles per day on average? Assume that the variable is normally distributed. Use the P-value method with a graphing calculator. Part 1 of 4 (a) State the hypotheses and identify the claim. H0: (Choose one) H1: (Choose one) This hypothesis test is a (Choose one) test.

Daily Driving The average number of miles a person drives per day is 24. A researcher wishes to see if people over age 60 drive less than 24 miles per day. She selects a random sample of 35 drivers over the age of 60 and finds that the mean number of miles driven is 23.2. The population standard deviation is 3 miles. At α=0.01, is there sufficient evidence that those drivers over 60 years old drive less than 24 miles per day on average? Assume that the variable is normally distributed. Use the P-value method with a graphing calculator.

Part 1 of 4
(a) State the hypotheses and identify the claim.

H0: (Choose one)
H1: (Choose one)

This hypothesis test is a (Choose one) test.

Transcript text: Daily Driving The average number of miles a person drives per day is 24. A researcher wishes to see if people over age 60 drive less than 24 miles per day. She selects a random sample of 35 drivers over the age of 60 and finds that the mean number of miles driven is 23.2. The population standard deviation is 3 miles. At $\alpha=0.01$, is there sufficient evidence that those drivers over 60 years old drive less than 24 miles per day on average? Assume that the variable is normally distributed. Use the $P$-value method with a graphing calculator. Part 1 of 4 (a) State the hypotheses and identify the claim. \[ \begin{array}{l} H_{0}: \square(\text { Choose one }) \nabla \\ H_{1}: \square(\text { (Choose one } \nabla \end{array} \] This hypothesis test is a $\square$ (Choose one) test.

Solution

Solution Steps

Step 1: State the Hypotheses

The hypotheses for this test are defined as follows: \[ \begin{align_} H_0: & \quad \mu \geq 24 \quad \text{(Null hypothesis)} \\ H_1: & \quad \mu < 24 \quad \text{(Alternative hypothesis)} \end{align_} \] This is a left-tailed test.

Step 2: Calculate the Standard Error

The standard error $SE$ is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{3}{\sqrt{35}} \approx 0.5071 \]

Step 3: Calculate the Test Statistic

The test statistic $Z$ is calculated using the formula: \[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{23.2 - 24}{0.5071} \approx -1.5776 \]

Step 4: Calculate the P-value

For a left-tailed test, the P-value is given by: \[ P = T(z) \approx 0.0573 \]

Step 5: Decision Rule

The significance level is set at $\alpha = 0.01$. Since the P-value $0.0573$ is greater than $\alpha$, we fail to reject the null hypothesis.

Final Answer

There is not sufficient evidence to support the claim that people over age 60 drive less than 24 miles per day on average. Thus, the conclusion is: \[ \boxed{\text{Fail to reject } H_0} \]

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