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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3:09 minutes

Problem 6.1.41

Textbook Question

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

Verified step by step guidance

Step 1: Identify the context of the problem. The question is asking whether the population mean (μ) could be greater than $70. This typically involves hypothesis testing or confidence interval analysis.

Step 2: Determine the statistical method to use. If the problem provides sample data (e.g., sample mean, standard deviation, and sample size), you can construct a confidence interval for the population mean or perform a hypothesis test.

Step 3: If constructing a confidence interval, calculate the margin of error using the formula:

E = z

_α/2 ⋅ (σ/n), where

z

_α/2 is the critical value,

σ

is the population standard deviation (or sample standard deviation if

σ

is unknown), and

n

is the sample size.

Step 4: If performing a hypothesis test, set up the null hypothesis

H_{0} : μ ≤ 70

and the alternative hypothesis

H_{a} : μ > 70

. Use the test statistic formula:

z = (x̄ - μ ₀) / (σ / \sqrt{n})

, where

x̄

is the sample mean,

μ ₀

is the hypothesized mean, and

σ

is the standard deviation.

Step 5: Interpret the results. For a confidence interval, check if $70 is within the interval. If it is not, it suggests the population mean could be greater than $70. For a hypothesis test, compare the p-value to the significance level (e.g., 0.05). If the p-value is small, reject the null hypothesis, indicating that the population mean is likely greater than $70.

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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 population, representing a central point around which data points are distributed. It is a key parameter in statistics, often denoted by the symbol μ. Understanding the population mean is crucial for making inferences about the entire population based on sample data.

Hypothesis Testing

Hypothesis testing is a statistical method used to determine whether there is enough evidence in a sample to support a specific claim about a population parameter. In this context, one might test the hypothesis that the population mean is greater than $70. This involves comparing sample data against a null hypothesis to assess the likelihood of observing the data if the null hypothesis is true.

Confidence Intervals

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. For example, if a confidence interval for the population mean does not include $70, it suggests that the mean is unlikely to be greater than this value. Understanding confidence intervals helps in assessing the precision and reliability of estimates.

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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 = 0.98, 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 35, would it be unusual for the population mean to be over $1500? 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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Textbook Question

Determining a Minimum Sample Size Determine the minimum sample size required when you want to be 99% confident that the sample mean is within two units of the population mean and σ = 1.4. Assume the population is normally distributed.

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