Question

use the accompanying 200 los angeles commute times to test the claim that the mean los angeles commute time is less than 33 minutes. assume that a simple random sample has been selected. use a 0.10 significance level. compare the sample mean to the claimed mean of 33 minutes. is the difference between these two values statistically significant? click the icon to view the los angeles commute times. identify the null and alternative hypotheses. ( h_0: mu = 33 ) ( h_1: mu < 33 ) (type integers or decimals. do not round.) identify the test statistic. -1.40 (round to two decimal places as needed.) identify the p - value 0.081 (round to three decimal places as needed.) state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. reject the null hypothesis. there is sufficient evidence at the 0.10 significance level to support the claim that the mean los angeles commute time is less than 33 minutes. the sample mean is minute(s) the claimed mean of 33 minutes; this difference statistically significant by the standard of this test. (round to one decimal place as needed.)

Explanation:

Step1: Hypotheses

The null hypothesis \(H_0:\mu = 33\) (claiming no difference from the mean of 33 minutes). The alternative hypothesis \(H_1:\mu<33\) (claiming the mean is less than 33 minutes).

Step2: Test Statistic

Given the test statistic \(t=- 1.40\) (calculated using the formula \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(\mu = 33\) is the population mean, \(s\) is the sample standard deviation, and \(n\) is the sample size).

Step3: P - value

The P - value is \(0.081\). Since the test is left - tailed (because \(H_1:\mu<33\)), the P - value is the probability of getting a test statistic as extreme or more extreme than the observed one under \(H_0\).

Step4: Conclusion

Since the P - value (\(0.081\)) is less than the significance level \(\alpha = 0.10\), we reject the null hypothesis. There is sufficient evidence at the \(0.10\) significance level to support the claim that the mean Los Angeles commute time is less than 33 minutes.

For the last part:
Let's assume the sample mean is calculated (using the data in the "Los Angeles commute times" which is not shown here, but for the sake of illustration, if we assume some value). Suppose the sample mean \(\bar{x}\) is calculated. The difference between the sample mean and the claimed mean is \(\bar{x}-33\). Since we rejected \(H_0\), the difference is statistically significant.

Answer:

Reject the null hypothesis. There is sufficient evidence at the \(0.10\) significance level to support the claim that the mean Los Angeles commute time is less than 33 minutes.
If we assume (for example) the sample mean \(\bar{x}=31.5\) (this value is assumed as the data is not given), the sample mean is \(1.5\) minute(s) less than the claimed mean of 33 minutes, and the difference is statistically significant by the standard of this test.