Constructing a Confidence Interval In Exercises 25–28, use the data set to (a) find the sample mean. Assume the population is normally distributed.
SAT Scores The SAT scores of 12 randomly selected high school seniors

Verified step by step guidance

1. Step 1: Identify the data set provided. The SAT scores of 12 randomly selected high school seniors are: 1130, 1290, 1010, 1320, 950, 1250, 1340, 1100, 1260, 1180, 1470, and 920.
2. Step 2: Calculate the sample mean. To do this, sum all the SAT scores and divide by the total number of scores (n = 12). Use the formula: $x=\frac{\sum x}{}$_in, where $\sum x$_i is the sum of all scores and $n$ is the sample size.
3. Step 3: Assume the population is normally distributed, which is a key condition for constructing a confidence interval.
4. Step 4: To construct the confidence interval, calculate the sample standard deviation using the formula: $s=\sqrt{\frac{\sum (x}{}}$_i-x)2n-1, where $x$ is the sample mean and $n$ is the sample size.
5. Step 5: Use the sample mean and standard deviation to calculate the confidence interval. For a 95% confidence level, use the formula: $x\pm t\bullet \frac{s}{\sqrt{n}}$, where $t$ is the t-score corresponding to the confidence level and degrees of freedom (df = n - 1).

Key Concepts

Sample Mean

The sample mean is the average of a set of values, calculated by summing all the observations and dividing by the number of observations. In this context, it represents the average SAT score of the 12 randomly selected high school seniors. It is a key statistic used to summarize data and is foundational for further statistical analysis, such as constructing confidence intervals.

Confidence Interval

A confidence interval is a range of values, derived from a data set, that is likely to contain the true population parameter with a specified level of confidence, typically 95% or 99%. It provides an estimate of uncertainty around the sample mean. Understanding how to construct and interpret confidence intervals is crucial for making inferences about the population based on sample data.

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. Many statistical methods, including the construction of confidence intervals, assume that the underlying population is normally distributed. This assumption is important for the validity of the results obtained from the sample data.