state gasoline taxes a random sample of state gasoline taxes (in cents) is shown here for 8 states. round…

Explanation:

Step1: List the sample data and count observations

The sample data is: $44.8, 58.7, 58.3, 44, 43.9, 58.7, 58.6, 46.7$. The sample size is $n = 8$.

Step2: Calculate the sample mean

$$\bar{x} = \frac{\sum x_i}{n} = \frac{44.8 + 58.7 + 58.3 + 44 + 43.9 + 58.7 + 58.6 + 46.7}{8} = \frac{413.7}{8} = 51.7125$$

Step3: Calculate the sample standard deviation

$$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \approx \sqrt{\frac{355.14875}{7}} \approx 6.7136$$

Step4: Determine the critical t-value

For $80%$ confidence and $df = n - 1 = 7$, the critical value $t_{\alpha/2}$ is: $$t_{0.10, 7} \approx 1.415$$

Step5: Calculate the margin of error

$$E = t_{\alpha/2} \cdot \left(\frac{s}{\sqrt{n}}\right) = 1.415 \cdot \left(\frac{6.7136}{\sqrt{8}}\right) \approx 1.415 \cdot 2.3736 \approx 3.3587$$

Step6: Construct the confidence interval

$$\text{Lower limit} = \bar{x} - E = 51.7125 - 3.3587 = 48.3538$$ $$\text{Upper limit} = \bar{x} + E = 51.7125 + 3.3587 = 55.0712$$

Answer:

$48.35 < \mu < 55.07$