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Chapter 8: Problem 57

Critical values What critical value t* from Table B would you use for a confidence interval for the population mean in each of the following situations? (a) A 95% confidence interval based on n 10 observations. (b) A 99% confidence interval from an SRS of 20 observations.

Short Answer

Expert verified
(a) t* = 2.262; (b) t* = 2.861.

Step by step solution

01

Identify the Degrees of Freedom

For a confidence interval using the t-distribution, we identify the degrees of freedom (df), which is the sample size (n) minus 1. For (a), n = 10, so df = 10 - 1 = 9. For (b), n = 20, so df = 20 - 1 = 19.
02

Determine the Significance Level

The confidence level is complementary to the significance level (alpha, denoted as \( \alpha \)). For a 95% confidence level, \( \alpha = 1 - 0.95 = 0.05 \). For a 99% confidence level, \( \alpha = 1 - 0.99 = 0.01 \).
03

Look Up Critical Value in t-Table

Consult the t-distribution table (Table B) using the calculated degrees of freedom and the corresponding significance level. - For (a), \( \alpha = 0.05 \) with df = 9, the critical value is \( t* = 2.262 \) for a two-tailed test.- For (b), \( \alpha = 0.01 \) with df = 19, the critical value is \( t* = 2.861 \) for a two-tailed test.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

t-distribution
The t-distribution is a probability distribution that is used when estimating population parameters, particularly when working with small sample sizes and the population variance is unknown. It resembles the normal distribution, but has thicker tails, which means there's a greater chance for extreme values. As sample size increases, the t-distribution approaches the normal distribution. It is a crucial tool in statistics for constructing confidence intervals and performing hypothesis tests when the sample size is small.
  • Used for small sample sizes
  • More spread with thicker tails compared to normal distribution
  • Converges to the normal distribution as sample size grows
Understanding the t-distribution helps determine how much risk or uncertainty is involved when making inferences about a population based on sample data. This distribution is indexed by degrees of freedom, which adds a degree of flexibility depending on how much data you have.
degrees of freedom
Degrees of freedom (df) refer to the number of independent values that a statistical calculation is based upon. They are essentially the number of values in the final calculation of a statistic that are free to vary. In the context of estimating means with a t-distribution, the degrees of freedom are calculated as the sample size minus one ( - 1).
  • Determines the shape of the t-distribution
  • df = sample size - 1
In our exercise, for (a) with 10 observations, df = 9. Meanwhile, for (b) with 20 observations, df = 19. The degrees of freedom affect the spread of the distribution. Fewer degrees of freedom result in a wider distribution, which means there's more variability and uncertainty in estimates.
critical values
Critical values are thresholds that determine the boundary or cut-off points for statistical significance within a given confidence level. In statistical tests, they originate from the t-distribution (or other probability distributions), and they help set the width of the confidence interval.
  • Define the range for confidence intervals
  • Derived from t-distribution tables
  • Depend on degrees of freedom and significance level
For a two-tailed test, as in our exercise, if the significance level is split into two tails of the distribution, the critical value identifies the t* score where the cumulative probability equals the confidence level. For (a), a 95% confidence interval with df = 9 results in a critical value of 2.262. For (b), a 99% confidence interval with df = 19 gives a critical value of 2.861. These critical values help establish the margins of error in confidence intervals.
significance level
The significance level (\( \alpha \)) is a probability measure that defines the cutoff threshold for determining statistical significance. It represents the likelihood of rejecting the null hypothesis when it is actually true (type I error). The significance level is complementary to the confidence level. For example, a 95% confidence level corresponds to a significance level of 0.05.
  • \( \alpha \) = 1 - confidence level
  • Determines risk of type I error
In our exercise:- For a 95% confidence interval, \( \alpha = 0.05 \).- For a 99% confidence interval, \( \alpha = 0.01 \).The significance level helps define how strict the criteria for statistical significance are. A lower \( \alpha \), as seen with 99% confidence, indicates more stringent criteria, reducing the probability of a type I error.

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