In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?


c=0.95, x̅=3.46, s=1.63, n=16

Verified step by step guidance

1. Step 1: Identify the type of distribution to use. Since the sample size (n = 16) is small and the population standard deviation is not provided, use the t-distribution to construct the confidence interval.
2. Step 2: Determine the degrees of freedom (df) for the t-distribution. The formula for degrees of freedom is df = n - 1. Here, df = 16 - 1 = 15.
3. Step 3: Find the critical t-value (t*) for a 95% confidence level and 15 degrees of freedom. Use a t-distribution table or statistical software to find this value.
4. Step 4: Calculate the margin of error (ME) using the formula: ME = t* × (s / √n), where s is the sample standard deviation, and n is the sample size.
5. Step 5: Construct the confidence interval using the formula: Confidence Interval = x̅ ± ME, where x̅ is the sample mean. This will give you the lower and upper bounds of the interval.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. For example, a 95% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 95% of those intervals would contain the true population mean.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. When constructing confidence intervals for the population mean, if the sample size is large (typically n > 30) or the population is normally distributed, the sample mean can be assumed to follow a normal distribution, allowing for the use of z-scores.

t-Distribution

The t-distribution is a type of probability distribution that is used when the sample size is small (n < 30) and the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, which provides a more accurate estimate of the confidence interval for the population mean in such cases, as it accounts for the increased variability in smaller samples.