To test H0: μ = 100 versus H1: μ ≠ 100, a simple random sample of size n = 23 is obtained from a population that is known to be normally distributed.


e. Construct a 99% confidence interval to test the hypothesis.

Verified step by step guidance

1. Identify the sample size \(n = 23\) and the population is normally distributed, which allows us to use the \(t\)-distribution for inference since the population standard deviation is unknown.
2. Determine the confidence level, which is 99%, so the significance level \(\alpha = 1 - 0.99 = 0.01\). Since this is a two-tailed test, split \(\alpha\) into two tails: \(\alpha/2 = 0.005\).
3. Find the critical value \(t_{\alpha/2, n-1}\) from the \(t\)-distribution table with degrees of freedom \(df = n - 1 = 22\) and tail probability \$0.005$.
4. Calculate the sample mean \(\bar{x}\) and the sample standard deviation \(s\) from the data (these values should be given or calculated from the sample).
5. Construct the 99% confidence interval for the population mean \(\mu\) using the formula: \(\displaystyle \bar{x} \pm t_{\alpha/2, n-1} \times \frac{s}{\sqrt{n}}\) This interval will help you test the hypothesis by checking if 100 lies within this interval.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis (H0) in favor of an alternative hypothesis (H1). It involves comparing sample data against a hypothesized population parameter, using significance levels to control error rates.

Confidence Interval

A confidence interval estimates a range of values within which the true population parameter is likely to fall, with a specified level of confidence (e.g., 99%). It provides an interval estimate rather than a single point, reflecting the uncertainty inherent in sampling.

t-Distribution and Small Sample Inference

When the sample size is small (n < 30) and the population standard deviation is unknown, the t-distribution is used instead of the normal distribution. It accounts for extra variability and is essential for constructing accurate confidence intervals and hypothesis tests with small samples.