Constructing a Confidence Interval In Exercises 25–28, use the da...

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1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 3m
- Introduction to Confidence Intervals
  15m
- Confidence Intervals for Population Mean
  48m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample1h 1m
- Steps in Hypothesis Testing
  1h 1m
10. Hypothesis Testing for Two Samples2h 8m
11. Correlation48m
- Scatterplots & Intro to Correlation
  26m
- Correlation Coefficient
  21m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

3. Describing Data Numerically

Mean

Statistics

3. Describing Data Numerically

Mean

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1:22 minutes

Problem 6.2.25a

Textbook Question

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

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.

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 = \sum x

_in, where

\sum x

_i is the sum of all scores and

n

is the sample size.

Step 3: Assume the population is normally distributed, which is a key condition for constructing a confidence interval.

Step 4: To construct the confidence interval, calculate the sample standard deviation using the formula:

s = \sqrt{\sum (x}

_i-x)2n-1, where

x

is the sample mean and

n

is the sample size.

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 ∙ \frac{s}{\sqrt{n}}

, where

t

is the t-score corresponding to the confidence level and degrees of freedom (df = n - 1).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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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