A local transportation official claims that the mean waiting time at the Bay Bridge toll booth is no more 

than 0.40 minutes. 33 randomly selected motorists were surveyed exiting the toll and it was found that 

the mean waiting time was 0.46 minutes with a standard deviation of 0.16 minutes. We will test the 

official's claim using a 0.01 level of significance.

To test the claim that the mean waiting time at the Bay Bridge toll booth is no more than 0.40 minutes, we can perform a hypothesis test using the following steps:

### Step 1: Set Up Hypotheses
- Null Hypothesis (\(H_0\)): \(\mu \leq 0.40\) (The mean waiting time is no more than 0.40 minutes)
- Alternative Hypothesis (\(H_a\)): \(\mu > 0.40\) (The mean waiting time is greater than 0.40 minutes)

### Step 2: Choose Significance Level
The significance level (\(\alpha\)) is given as 0.01.

### Step 3: Calculate the Test Statistic
Since the sample size is 33 (which is less than 30) and we do not know the population standard deviation, we will use a t-test.

1. **Sample Mean (\(\bar{x}\))**: 0.46 minutes
2. **Sample Standard Deviation (s)**: 0.16 minutes
3. **Sample Size (n)**: 33

The formula for the t-test statistic is:

\[
t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}
\]

Where:
- \(\mu_0 = 0.40\) (the value stated in the null hypothesis)
- \(\bar{x} = 0.46\)
- \(s = 0.16\)
- \(n = 33\)

Calculating \(t\):

\[
t = \frac{0.46 - 0.40}{0.16 / \sqrt{33}} \approx \frac{0.06}{0.0277} \approx 2.17
\]

### Step 4: Determine the Critical Value
For a one-tailed t-test at \(\alpha = 0.01\) with \(n - 1 = 32\) degrees of freedom, we can find the critical t-value from t-distribution tables or calculators. The critical t-value for \(df = 32\) at a significance level of 0.01 is approximately 2.457.

### Step 5: Decision Rule
- If the calculated \(t\) is greater than the critical \(t\), we reject the null hypothesis.

### Step 6: Compare and Conclude
- Calculated \(t\) = 2.17
- Critical \(t\) = 2.457

Since \(2.17 < 2.457\), we do not reject the null hypothesis.

### Conclusion
At the 0.01 level of significance, there is not enough evidence to support the claim that the mean waiting time at the Bay Bridge toll booth is greater than 0.40 minutes.

### Summary
1. The official's claim that the mean waiting time is no more than 0.40 minutes is not rejected based on the sample data.
2. The mean waiting time observed in the sample is 0.46 minutes, but it's not statistically significant at the 0.01 level.

Do you want more details on any part of this analysis or have specific questions? Here are some related questions you might consider:

1. What assumptions are made in this hypothesis test?
2. How would the conclusion change with a different significance level?
3. What are the implications of the test results for the local transportation officials?
4. How can the sample size affect the power of the test?
5. What would be the result if the sample mean was 0.39 minutes? 

**Tip:** Always check assumptions of normality and independence before conducting hypothesis tests, especially with smaller sample sizes.

A local transportation official claims that the mean waiting time at the Bay Bridge toll booth is no more than 0.40 minutes. 33 randomly selected motorists were surveyed exiting the toll and it was found that the mean waiting time was 0.46 minutes with a standard deviation of 0.16 minutes. We will test the official's claim using a 0.01 level of significance.

Perform a hypothesis test to verify a claim about the mean waiting time at the Bay Bridge toll booth. Using a sample mean of 0.46 minutes, standard deviation of 0.16 minutes, and a 0.01 significance level, this problem covers hypothesis testing concepts and the t-test formula, suitable for high school or college students.

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