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Ch. 6 - Confidence Intervals
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 6 - Confidence IntervalsProblem 6.2.33
Chapter 6, Problem 6.2.33

In Exercise 31, the population mean salary is \$67,319. Does the t-value fall between -t0.98 and t0.98? (Source: Salary.com)

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Step 1: Identify the given information. The population mean salary is \$67,319. The problem asks whether the t-value falls between -t0.98 and t0.98. This implies a two-tailed test with a confidence level of 98%.
Step 2: Determine the degrees of freedom (df). The degrees of freedom depend on the sample size (n). If the sample size is not provided, you cannot calculate the exact t-values. Ensure you know the sample size to proceed.
Step 3: Use a t-distribution table or statistical software to find the critical t-values for a 98% confidence level. For a two-tailed test, the critical t-values correspond to the upper and lower 1% tails of the distribution (since 100% - 98% = 2%, and 2%/2 = 1% in each tail).
Step 4: Calculate the t-value for the sample data using the formula: t = (x̄ - μ) / (s / √n), where x̄ is the sample mean, μ is the population mean (\$67,319), s is the sample standard deviation, and n is the sample size. Plug in the values to compute the t-value.
Step 5: Compare the calculated t-value to the critical t-values (-t0.98 and t0.98). If the calculated t-value falls within this range, it supports the null hypothesis. Otherwise, it suggests evidence against the null hypothesis.

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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. It is calculated by summing all the values and dividing by the number of values. In this context, the population mean salary of $67,319 represents the average salary of all individuals in the specified population, which is crucial for understanding the distribution of salaries.
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Population Standard Deviation Known

T-Value

The t-value is a statistic that measures the size of the difference relative to the variation in your sample data. It is used in hypothesis testing to determine if the means of two groups are significantly different from each other. In this case, the t-value helps assess whether the observed sample mean salary significantly deviates from the population mean.
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Critical Values: t-Distribution

Critical Values

Critical values are the threshold points that define the boundaries of the acceptance region in hypothesis testing. For a t-distribution, these values are determined based on the desired significance level and degrees of freedom. The range between -t0.98 and t0.98 indicates the values within which the t-value must fall to fail to reject the null hypothesis, suggesting no significant difference.
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Critical Values: t-Distribution
Related Practice
Textbook Question

True or False? In Exercises 1 and 2, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

The point estimate for the population proportion of failures is 1-p^

Textbook Question

LGBT Identification In a survey of 15,349 U.S. adults, 860 identify as lesbian, gay, bisexual, or transgender. Construct a 95% confidence interval for the population proportion of U.S. adults who identify as lesbian, gay, bisexual, or transgender. (Adapted from Gallup)

Textbook Question

In Exercises 13–16, find the margin of error for the values of c, σ and n.

c = 0.975, σ = 4.6, n = 100

Textbook Question

When estimating the population mean, why not construct a 99% confidence interval every time?

Textbook Question

In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.

c = 0.80, σ = 4.1, E = 2.

Textbook Question

Matching In Exercises 17–20, match the level of confidence c with the appropriate confidence interval. Assume each confidence interval is constructed for the same sample statistics.

c = 0.98