The department of transportation has mandated that the average speed of cars no interstate highways be no more than 70 km per hours in order. To check that the people follow it or not, a researcher took a random sample of 186 cars and found that the average speed was 68 km per hours with a standard deviation 0.6 km per hours.

(a) Find the standard error of the mean

(b) Construct the interval around the sample mean that would contain the population mean 95% of the time.

(c) If the researcher wants to test that the true mean speed on its highways is 70 km per hours or less with 95% confidence then

(i) State null and alternative hypotheses

(ii) Name the test which is suitable in this situation and why?

(iii)Calculate the value of test statistic and critical value.

(iv)Draw the conclusion on the basis of the applied test.

(a) Find the standard error of the mean:

The standard error of the mean (SEM) is calculated using the formula:

Standard Error=sn\text{Standard Error} = \frac{s}{\sqrt{n}}

Where:

• ss is the standard deviation of the sample,
• nn is the sample size.

From the problem, we know:

• s=0.6 km/hs = 0.6 \, \text{km/h},
• n=186n = 186.

Substituting these values into the formula:

Standard Error=0.6186≈0.613.65≈0.0439 km/h.\text{Standard Error} = \frac{0.6}{\sqrt{186}} \approx \frac{0.6}{13.65} \approx 0.0439 \, \text{km/h}.

Thus, the standard error of the mean is approximately 0.0439 km/h.

(b) Construct the interval around the sample mean that would contain the population mean 95% of ______ _____ _______ ____ __________ ________.
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