Question

Unhealthy Days in Cities The number of unhealthy days based on the AQI (Air Quality Index) for a random sample of metropolitan areas is shown. Round the
sample statistics and final answers to at least one decimal place.
275, 50, 6, 15, 3, 23, 21, 14, 5
Construct a 90% confidence interval based on the data. Assume the variable is normally distributed.

Answer 1

Using the t-distribution, the 90% confidence interval based on the data is (0, 99.79).

The sample mean is:

$\overline{x} = \frac{275 + 50 + 6 + 15 + 3 + 23 + 21 + 14 + 5}{9} = 45.8$

The sample standard deviation is:

$s = \sqrt{\frac{(275-45.8)^2+(50-45.8)^2+(6-45.8)^2+...+(14-45.8)^2+(5-45.8)^2}{8}} = 87.1$

We have the standard deviation for the sample, which is why the t-distribution is used to solve this question.

The confidence interval is:

$\overline{x} \pm t\frac{s}{\sqrt{n}}$

The critical value, using a t-distribution calculator, for a two-tailed 90% confidence interval, with 9 - 1 = 8 df, is t = 1.8595.

Then, the interval is:

$\overline{x} - t\frac{s}{\sqrt{n}} = 45.8 - 1.8595\frac{87.1}{\sqrt{9}} = -8.19$

$\overline{x} + t\frac{s}{\sqrt{n}} = 45.8 + 1.8595\frac{87.1}{\sqrt{9}} = 99.79$

The number of days cannot be negative, thus we consider the lower bound 0 and the 90% confidence interval based on the data is (0, 99.79).

A similar problem is given at https://brainly.com/question/15180581