Calculate CI and Test Correlation Under Normality
Statistical Inference and Distribution Analysis
This problem requires you to apply fundamental statistical inference techniques, assuming standard parametric conditions, specifically normality. For each part, you must:
- Show the exact formulas used.
- Identify degrees of freedom where relevant.
- Provide clear numeric answers.
Part (a): 95% Confidence Interval for a Normal Mean (Population Variance Unknown)
You are given a a simple random sample of size n = 64 drawn from a normal population with unknown variance. The sample mean is x̄ = 102, and the sample standard deviation is s = 16.
Compute a 95% confidence interval for the population mean μ. Explicitly state the formula used and the degrees of freedom for the t-critical value.
Part (b): Test Significance of a Pearson Correlation
You have n = 100 observations for two variables, X and Y, yielding a sample Pearson correlation coefficient r = 0.25.
Perform a hypothesis test for the population correlation coefficient ρ, with the null hypothesis H₀: ρ = 0 against the alternative hypothesis H₁: ρ ≠ 0, using a significance level α = 0.05.
Your answer should include the test statistic, its degrees of freedom, the two-sided p-value, and your final conclusion.
Part (c): Upper 2.5% Threshold of a Normal Process
Consider a process that is normally distributed with a mean μ = 50 and a standard deviation σ = 5.
Determine the threshold value t such that only the top 2.5% of items from this process exceed t. Interpret the meaning of this threshold t within the context of the problem.
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Drawing robust and reliable conclusions about population parameters from limited sample data, while correctly accounting for inherent uncertainty and underlying distributional assumptions.
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Calculate CI and Test Correlation Under Normality
Statistical Inference and Distribution Analysis
This problem requires you to apply fundamental statistical inference techniques, assuming standard parametric conditions, specifically normality. For each part, you must:
- Show the exact formulas used.
- Identify degrees of freedom where relevant.
- Provide clear numeric answers.
Part (a): 95% Confidence Interval for a Normal Mean (Population Variance Unknown)
You are given a a simple random sample of size n = 64 drawn from a normal population with unknown variance. The sample mean is x̄ = 102, and the sample standard deviation is s = 16.
Compute a 95% confidence interval for the population mean μ. Explicitly state the formula used and the degrees of freedom for the t-critical value.
Part (b): Test Significance of a Pearson Correlation
You have n = 100 observations for two variables, X and Y, yielding a sample Pearson correlation coefficient r = 0.25.
Perform a hypothesis test for the population correlation coefficient ρ, with the null hypothesis H₀: ρ = 0 against the alternative hypothesis H₁: ρ ≠ 0, using a significance level α = 0.05.
Your answer should include the test statistic, its degrees of freedom, the two-sided p-value, and your final conclusion.
Part (c): Upper 2.5% Threshold of a Normal Process
Consider a process that is normally distributed with a mean μ = 50 and a standard deviation σ = 5.
Determine the threshold value t such that only the top 2.5% of items from this process exceed t. Interpret the meaning of this threshold t within the context of the problem.