Math Snap

STEP 1

1. The data represents the number of unhealthy days based on the AQI for a sample of metropolitan areas.
2. The variable is normally distributed.
3. We are constructing a $98\%$ confidence interval for the mean.
4. The sample size is $n = 8$.

STEP 2

1. Calculate the sample mean ($\bar{x}$).
2. Calculate the sample standard deviation ($s$).
3. Determine the critical value ($t^*$) for a $98\%$ confidence level.
4. Calculate the margin of error.
5. Construct the confidence interval.

STEP 3

Calculate the sample mean ($\bar{x}$):
$$\bar{x} = \frac{29 + 23 + 10 + 32 + 24 + 0 + 5 + 27}{8}$$ $$\bar{x} = \frac{150}{8} = 18.75$$

STEP 4

Calculate the sample standard deviation ($s$):
First, find the deviations from the mean and square them:
$$(29 - 18.75)^2, (23 - 18.75)^2, (10 - 18.75)^2, (32 - 18.75)^2, (24 - 18.75)^2, (0 - 18.75)^2, (5 - 18.75)^2, (27 - 18.75)^2$$ Calculate each squared deviation:
$$104.0625, 18.0625, 76.5625, 174.0625, 27.5625, 351.5625, 189.0625, 67.5625$$ Sum the squared deviations and divide by $n-1$:
$$s^2 = \frac{104.0625 + 18.0625 + 76.5625 + 174.0625 + 27.5625 + 351.5625 + 189.0625 + 67.5625}{7}$$ $$s^2 = \frac{1008.5}{7} = 144.0714$$ Calculate the standard deviation:
$$s = \sqrt{144.0714} \approx 12.0$$

STEP 5

Determine the critical value ($t^<em>$) for a $98\%$ confidence level with $n-1 = 7$ degrees of freedom. Using a $t$-distribution table or calculator, find $t^</em> \approx 2.998$.

STEP 6

Calculate the margin of error:
$$\text{Margin of Error} = t^* \times \frac{s}{\sqrt{n}}$$ $$\text{Margin of Error} = 2.998 \times \frac{12.0}{\sqrt{8}}$$ $$\text{Margin of Error} \approx 2.998 \times 4.24 \approx 12.7$$

Construct the confidence interval:
$$\bar{x} - \text{Margin of Error} < \mu < \bar{x} + \text{Margin of Error}$$ $$18.75 - 12.7 < \mu < 18.75 + 12.7$$ $$6.05 < \mu < 31.45$$ The $98\%$ confidence interval for the mean number of unhealthy days is:
$$\boxed{6.05 < \mu < 31.45}$$