In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.
c = 0.95, n = 13

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with finding the critical values for a given confidence level (c = 0.95) and sample size (n = 13). Critical values are used in hypothesis testing and confidence intervals to determine the range within which the true population parameter lies.
2. Step 2: Identify the appropriate distribution. Since the sample size is small (n < 30), and assuming the population standard deviation is unknown, you will use the t-distribution to find the critical values.
3. Step 3: Determine the degrees of freedom (df). The degrees of freedom for the t-distribution is calculated as: $\mathrm{df}=n-1$. For this problem, $\mathrm{df}=13-1=12$.
4. Step 4: Use the confidence level to find the critical values. The confidence level c = 0.95 corresponds to a two-tailed test, meaning the area in each tail is $\frac{1}{2}(1-c)$. For c = 0.95, the area in each tail is $\frac{1}{2}(1-0.95)=0.025$. The critical values correspond to the t-scores where the cumulative probability equals 0.025 in the lower tail and 0.975 in the upper tail.
5. Step 5: Look up the t-scores in a t-distribution table or use statistical software. Using the degrees of freedom (df = 12) and the cumulative probabilities (0.025 and 0.975), find the critical values. These values will be symmetric around zero, with one positive and one negative.

Key Concepts

Critical Values

Critical values are the points on the scale of the test statistic that define the boundaries for rejecting the null hypothesis. They are determined based on the desired level of confidence and the distribution of the test statistic. For example, in a normal distribution, critical values correspond to specific z-scores that capture the central area of the distribution, reflecting the confidence level.

Level of Confidence (c)

The level of confidence, denoted as 'c', represents the probability that the confidence interval will contain the true population parameter. A common level of confidence is 95%, which implies that if we were to take many samples and construct confidence intervals, approximately 95% of those intervals would contain the true parameter. This level influences the width of the confidence interval.

Sample Size (n)

Sample size, denoted as 'n', refers to the number of observations or data points collected in a study. The sample size affects the precision of the estimates and the width of the confidence intervals; larger sample sizes generally lead to more reliable estimates and narrower intervals. In this case, with n = 13, the sample size is relatively small, which may impact the critical values derived from the statistical distribution.