Question

One of the two fire stations in a certain town responds to calls in the northern half of the town, and the other fire station responds to calls in the southern half of the town. The following is a list of response times (in minutes) for both of the fire stations (this data will be used for several problems). Both samples may be regarded as simple random samples from approximately normal populations so that the t- procedures are safe to use. Northern: 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 7, 7, 8, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 12 Sum = 192 Sum of Squared Deviations = 197.2 Southern: 4,4,4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 12, 12, 12, 12, 13 Sum = 225 Sum of Squared Deviations = 231.5 Find and interpret a 95% confidence interval for the mean response time of the fire station that responds to calls in the northern part of town. Fill in blank 1 to report the bounds of the 95% CI. Enter your answers as lower bound upper bound with no additional spaces and rounding bounds to two decimals.

Answer 1

The sample mean response time in the northern part of the town is

$\overline{x} = \frac{192}{30} = 6.4$

The sample standard deviation of the response time in the northern part of the town is

$\bg_white s = \sqrt\frac{197.2}{30-1} = 2.608$

The degree of freedom is = 30 - 1 = 29

The critical value of t is = 2.045

The 95% confidence interval for the mean response time in the northern part of the town is

$= 6.4 \pm 2.045 * \frac{2.608}{\sqrt{30}}$

The 95% confidence interval for the mean response time in the northern part of the town is

$= 5.43 < \mu < 7.37$

On plotting the points, we get the following curve

Normal Curve 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 N 0 4 6 8 10 12 14

The curve is very slightly skewed, and hence we can use the confidence interval. The sample size of at least 30 is required. Here, that criteria is satisfied, and the curve looks approximately like a normal curve, hence it would be okay to proceed like that it calculating the confidence inetrval.

The correct option is (3).

First option is incorrect because the curve is almost symmetric and the sample size is also sufficiently large.

Second option is incorrect because the curve is not extremely skewed.

Fourth option is incorrect because the t-distribution is not always safe to use. It is not safe to use if the distribution is extremely skewed.

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