Questions: Test 3 Question 6 of 20 (10 points) Question Attempt: 1 of 1 (a) Explain why it is necessary to check whether the population is approximately normal before constructing a confic It is necessary to check whether the population is approximately normal because the sample size is less than or equal to 30 Part: 1 / 3 Part 2 of 3 (b) Following is a dotplot of these data. Is it reasonable to assume that the population is approximately normal? 100 150 200 250 300 350 400 It (Choose one) reasonable to assume that the population is approximately normal.

Solution Steps

It is necessary to check whether the population is approximately normal because the sample size is less than or equal to 30. This is important for the validity of statistical methods that assume normality, particularly when constructing confidence intervals.

Based on the provided data, it is reasonable to assume that the population is approximately normal. This assumption is crucial for the subsequent analysis.

To calculate the confidence interval for the mean of a single population with unknown variance and a small sample size, we use the formula:

\[ \bar{x} \pm t \frac{s}{\sqrt{n}} \]

Where:

• \(\bar{x} = 250.0\) (sample mean)
• \(t = 2.45\) (t-value for 95% confidence level with \(n-1 = 6\) degrees of freedom)
• \(s = 108.01\) (sample standard deviation)
• \(n = 7\) (sample size)

Substituting the values into the formula gives:

\[ 250.0 \pm 2.45 \cdot \frac{108.01}{\sqrt{7}} = 250.0 \pm 2.45 \cdot 40.96 \]

Calculating the margin of error:

\[ 2.45 \cdot 40.96 \approx 100.89 \]

Thus, the confidence interval is:

\[ (250.0 - 100.89, 250.0 + 100.89) = (149.11, 350.89) \]

Final Answer