The data set represents the scores of 12 randomly selected students on the SAT Physics Subject Test. Assume the population test scores are normally distributed and the population standard deviation is 108. (Adapted from The College Board)

b. Construct a 90% confidence interval for the population mean. Interpret the results.

Verified step by step guidance

1. Step 1: Calculate the sample mean (x̄) using the provided data set. Add all the scores together and divide by the number of scores (12). The formula is x̄ = (Σx) / n.
2. Step 2: Identify the population standard deviation (σ), which is given as 108, and the sample size (n), which is 12.
3. Step 3: Determine the z-value corresponding to a 90% confidence level. For a 90% confidence interval, the z-value is approximately 1.645 (from the standard normal distribution table).
4. Step 4: Calculate the margin of error (E) using the formula E = z * (σ / √n). Substitute the values for z, σ, and n into the formula.
5. Step 5: Construct the confidence interval using the formula: Confidence Interval = x̄ ± E. Interpret the results by explaining that the interval provides a range within which the true population mean is likely to fall with 90% confidence.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from a data set, 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

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In this context, the assumption that the population test scores are normally distributed allows for the use of specific statistical methods, such as constructing confidence intervals.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In this case, the population standard deviation of 108 indicates how much the SAT Physics Subject Test scores deviate from the mean score. It is crucial for calculating the margin of error when constructing the confidence interval.