Table of contents

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 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

Statistics

7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

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3:43 minutes

Problem 6.1.39

Textbook Question

In Exercise 35, would it be unusual for the population mean to be over $1500? Explain.

Verified step by step guidance

Step 1: Identify the key components of the problem. The question is asking whether it would be unusual for the population mean to exceed $1500. To determine this, we need to assess the probability of the population mean being greater than $1500 using the concept of z-scores and the standard normal distribution.

Step 2: Recall the formula for the z-score for a population mean:

z = (μ - μ

₀) / (σ / n), where

μ

is the population mean,

μ

₀ is the hypothesized mean (in this case, $1500),

σ

is the population standard deviation, and

n

is the sample size.

Step 3: Substitute the given values into the z-score formula. If the problem does not provide the population standard deviation (

σ

) or sample size (

n

), you will need to refer to Exercise 35 for these details.

Step 4: Use the z-score obtained to find the corresponding probability from the standard normal distribution table. This will give you the likelihood of the population mean being less than or equal to $1500. To find the probability of the mean being greater than $1500, subtract this value from 1.

Step 5: Interpret the result. If the probability of the population mean being greater than $1500 is very small (commonly less than 0.05), it would be considered unusual for the population mean to exceed $1500. Otherwise, it would not be unusual.

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Key Concepts

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

Population Mean

The population mean is the average of all values in a given population. It is a measure of central tendency that provides insight into the overall characteristics of the population. Understanding the population mean is crucial for determining how individual data points relate to the average, which can help assess whether a specific value, like $1500, is typical or unusual.

Statistical Significance

Statistical significance refers to the likelihood that a result or relationship observed in data is not due to random chance. In the context of the population mean, determining whether a mean over $1500 is unusual involves assessing how far this value is from the expected mean and whether this deviation is statistically significant, often using hypothesis testing.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. Many statistical analyses assume normality, and understanding this concept helps in evaluating whether a population mean over $1500 falls within a typical range or is an outlier, based on the distribution of the data.

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Related Practice

Textbook Question

In Exercises 13–16, (a) find the margin of error for the values of c, s, and n, and (b) construct the confidence interval for using the t-distribution. Assume the population is normally distributed.

c = 0.90, s = 25.6, n = 16, xbar = 72.1

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Textbook Question

c = 0.98, s = 0.9, n = 12, xbar = 6.8

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Textbook Question

c = 0.99, s = 16.5, n = 20, xbar = 25.2

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Textbook Question

Finding the Margin of Error In Exercises 33 and 34, use the confidence interval to find the estimated margin of error. Then find the sample mean. Book Prices A store manager reports a confidence interval of (244.07, 280.97) when estimating the mean price (in dollars) for the population of textbooks.

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Textbook Question

In Exercise 37, does it seem likely that the population mean could be greater than $70? Explain.

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Textbook Question

When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.

a. Increase in the level of confidence

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Textbook Question

When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.

b. Increase in the sample size

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Textbook Question

When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.

c. Increase in the population standard deviation

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