Question

An automatic merge system has been installed at the entrance ramp to a major highway. Prior to the installation of the system, investigators found the average stress level of drivers to be 8.2 on a 10-point scale. After installation, a sample of 50 drivers showed a mean stress level of 7.6 with a standard deviation of 1.8. Conduct a hypothesis test using a significance level of 0.01 to determine if the average stress at peak hours for drivers under the new system is less than 8.2.

Answer 1

The average stress at peak hours for drivers under the new system is not less than 8.2.

Explanation:

In this case e need to determine whether the the average stress at peak hours for drivers under the new system is less than 8.2.

Prior to the installation of the automatic merge system, investigators found the average stress level of drivers to be 8.2 on a 10-point scale.

The hypothesis can be defined as follows:

H₀: The average stress at peak hours for drivers under the new system is not less than 8.2, i.e. μ ≥ 8.2.

Hₐ: The average stress at peak hours for drivers under the new system is less than 8.2, i.e. μ < 8.2.

The information provided is:

`n=50\\\bar x=7.6\\s=1.8\\\alpha =0.01`

As the population standard deviation is not provided, we will use a t-test for single mean.

Compute the test statistic value as follows:

`t=(\bar x-\mu)/(s/√(n))=(7.6-8.2)/(1.8/√(50))=-2.36`

The test statistic value is -2.36.

Decision rule:

If the p-value of the test is less than the significance level then the null hypothesis will be rejected.

Compute the p-value for the one-tailed t-test as follows:

`p-value=P(t_(\alpha, n-1)<t)\\=P(t_(0.01, 49)<t)\\=P(t_(0.01, 49)<-2.36)\\=P(t_(0.01, 49)>2.36)\\=0.011`

*Use a t-table for the probability.

The p-value of the test is 0.011.

The p-value of the test is greater than the significance level. The null hypothesis will not be rejected.

Thus, it can be concluded that the average stress at peak hours for drivers under the new system is not less than 8.2.