Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.Q.5

Chapter 6, Problem 6.Q.5

You research the salaries of senior-level civil engineers and find that the population mean is \$131,935. In Exercise 4, does the t-value fall between -t0.95 and t0.95?

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Identify the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis typically states that there is no significant difference, while the alternative hypothesis suggests there is a difference.

Determine the sample mean, sample size, and sample standard deviation from Exercise 4 (these values are necessary to calculate the t-value).

Calculate the t-value using the formula:

t = \frac{(x_{-} μ)}{\frac{s}{\sqrt{n}}}

, where x is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.

Find the critical t-values for a 95% confidence level (t0.95 and -t0.95) using a t-distribution table or statistical software. The degrees of freedom (df) are calculated as

df = n - 1

Compare the calculated t-value to the critical t-values. If the t-value falls between -t0.95 and t0.95, the null hypothesis is not rejected; otherwise, it is rejected.

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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 a set of values in a complete population. In this context, it represents the average salary of all senior-level civil engineers, which is $131,935. Understanding the population mean is crucial for making inferences about the data and comparing it to sample statistics.

t-Distribution

The t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution but with heavier tails. It is used in statistics when the sample size is small or the population standard deviation is unknown. The t-values, such as -t0.95 and t0.95, represent critical values that help determine the range of values for hypothesis testing.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis and an alternative hypothesis, then using test statistics (like t-values) to determine whether to reject the null hypothesis. In this case, checking if the t-value falls between -t0.95 and t0.95 helps assess the significance of the sample mean in relation to the population mean.

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06:21

Step 1: Write Hypotheses

Related Practice

Textbook Question

The data set represents the amounts of time (in minutes) spent checking email for a random sample of employees at a company.

c. Repeat part (b), assuming σ = 3.5 minutes. Compare the results.

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Textbook Question

[APPLET] The winning times (in hours) for a sample of 20 randomly selected Boston Marathon Women’s Open Division champions from 1980 to 2019 are shown in the table at the left. Assume the population standard deviation is 0.068 hour. (Source: Boston Athletic Association)

a. Find the point estimate of the population mean.

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Textbook Question

b. Find the margin of error for a 95% confidence level.

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Textbook Question

In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.

c = 0.90, n = 16

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Textbook Question

In a random sample of 12 senior-level civil engineers, the mean annual earnings were \$133,326 and the standard deviation was \$36,729. Assume the annual earnings are normally distributed and construct a 95% confidence interval for the population mean annual earnings for senior-level civil engineers. Interpret the results. (Adapted from Salary.com)

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Textbook Question

d. Does it seem likely that the population mean could be greater than 2.52 hours? Explain.

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