Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problema 6.2.33

Capitolo 6, Problema 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 -t_0.98 and t_0.98. 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 (-t_0.98 and t_0.98). 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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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

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