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


c=0.90, x̅=8.21, σ=0.62, n=8

Verified step by step guidance

1. Step 1: Identify the given values in the problem. The confidence level (c) is 0.90, the sample mean (x̅) is 8.21, the population standard deviation (σ) is 0.62, and the sample size (n) is 8.
2. Step 2: Determine the appropriate distribution to use. Since the population standard deviation (σ) is known and the sample size (n) is small (n < 30), use the standard normal (Z) distribution to construct the confidence interval.
3. Step 3: Find the critical value (Z*) corresponding to the confidence level of 0.90. To do this, calculate the area in each tail as (1 - c) / 2 = (1 - 0.90) / 2 = 0.05. Use a Z-table or statistical software to find the Z* value that leaves 0.05 in each tail.
4. Step 4: Calculate the margin of error (E) using the formula: $E\; =\; Z^*\; \backslash (\backslash cdot\backslash )\; \backslash (\backslash frac\{\backslash sigma\}\{\backslash sqrt\{n\}\backslash )\}$. Substitute the values for Z*, σ, and n into the formula.
5. Step 5: Construct the confidence interval for the population mean using the formula: $(x̅\; -\; E,\; x̅\; +\; E)$. Substitute the values for x̅ and E to find the lower and upper bounds of the confidence 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 90% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 90% 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, if the sample size is small (typically n < 30) and the population standard deviation is known, the t-distribution is often used instead of the normal distribution, especially when the underlying population is normally distributed.

Standard Error

The standard error (SE) measures the variability of the sample mean estimate of a population mean. It is calculated as the population standard deviation divided by the square root of the sample size (σ/√n). A smaller standard error indicates that the sample mean is a more accurate estimate of the population mean, which is crucial for constructing a reliable confidence interval.