To analyze the given SAT scores, let's go through each of the steps:

Given Data

SAT scores of 12 high school seniors: $1130, 1290, 1010, 1320, 950, 1250, 1340, 1100, 1260, 1180, 1470, 920$

Steps:

(a) Find the Sample Mean

The sample mean is calculated by summing all the data points and dividing by the number of data points.

(b) Find the Sample Standard Deviation

The sample standard deviation measures the spread of the data points around the sample mean. It’s calculated as: $s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}$ where $n$ is the sample size, $x_i$ are the individual data points, and $\bar{x}$ is the sample mean.

(c) Construct a 99% Confidence Interval for the Population Mean

Since we assume the population is normally distributed and the sample size is less than 30, we use the $t$-distribution to construct the confidence interval. The formula for the confidence interval is: $\bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$ where:

• $\bar{x}$ is the sample mean,
• $t_{\alpha/2}$ is the critical value from the $t$-distribution with $n - 1$ degrees of freedom for a 99% confidence level,
• $s$ is the sample standard deviation, and
• $n$ is the sample size.

Let's calculate each of these.Here are the results:

(a) Sample Mean

The sample mean of the SAT scores is: $\bar{x} = 1185.0$

(b) Sample Standard Deviation

The sample standard deviation is: $s = 168.06$

(c) 99% Confidence Interval for the Population Mean

The 99% confidence interval for the population mean is: $(1034.32, 1335.68)$

This interval means we are 99% confident that the true population mean of SAT scores lies between 1034.32 and 1335.68.

Would you like further details or have any questions?

Here are 5 related questions:

1. How would changing the confidence level affect the confidence interval?
2. What is the impact of increasing the sample size on the confidence interval?
3. How does using a $t$-distribution differ from a $z$-distribution in constructing confidence intervals?
4. What assumptions are necessary for the confidence interval to be valid?
5. How would outliers in the sample affect the standard deviation and confidence interval?

Tip: Remember, for small samples (n < 30), the $t$-distribution provides a more accurate confidence interval than the $z$-distribution.