Question 1205589: At the same distance with the same measuring device, B was measured 8 times. The average measurement result was 202m, with an average square deviation of s=0.7m.
1)Determine the confidence interval for the mathematical expectation of the measurement of distance B with a probability of 0.92.
2)Test the hypothesis: H0: σ^2 ≥ 0.65, H1: σ^2 < 0.65, with a significance level of 0.1.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! 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.
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