Question 1205589
Understanding the Problem

We have a sample of 8 measurements of distance B:

Sample size (n) = 8
Sample mean (x̄) = 202 meters
Sample standard deviation (s) = 0.7 meters
We're asked to:

Determine the 92% confidence interval for the population mean.
Test the hypothesis H₀: σ² ≥ 0.65 vs. H₁: σ² < 0.65 at a 0.1 significance level.
1. Confidence Interval for the Population Mean

We'll use a t-distribution since the population standard deviation (σ) is unknown.

Step 1: Determine the critical value (tα/2)

α = 1 - 0.92 = 0.08
Degrees of freedom (df) = n - 1 = 7
Using a t-table or statistical software, we find tα/2 ≈ 1.895
Step 2: Calculate the margin of error (ME)

ME = tα/2 * (s / √n) ≈ 1.895 * (0.7 / √8) ≈ 0.47
Step 3: Construct the confidence interval

Confidence interval = (x̄ - ME, x̄ + ME) ≈ (202 - 0.47, 202 + 0.47) ≈ (201.53, 202.47)
Therefore, we are 92% confident that the true population mean of distance B lies between 201.53 and 202.47 meters.

2. Hypothesis Testing for the Population Variance

We'll use a chi-square test for this hypothesis.

Step 1: Determine the critical value (χ²α)

α = 0.1
Degrees of freedom (df) = n - 1 = 7
Using a chi-square table or statistical software, we find χ²α ≈ 12.017
Step 2: Calculate the test statistic (χ²)

χ² = (n - 1) * s² / σ₀² = (7 * 0.7²) / 0.65 ≈ 5.6
Step 3: Make a decision

Since χ² (5.6) is less than χ²α (12.017), we fail to reject the null hypothesis H₀.
Therefore, we do not have enough evidence to conclude that the population variance is less than 0.65 at the 0.1 significance level.