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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2:09 minutes

Problem 6.2.33

Textbook Question

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

Verified step by step guidance

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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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.

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.

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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Textbook Question

The initial pressures for bicycle tires when first filled are normally distributed, with a mean of 70 pounds per square inch (psi) and a standard deviation of 1.2 psi.

b. A random sample of 15 tires is drawn from this population. What is the probability that the mean tire pressure of the sample is less than 69 psi?

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Textbook Question

Constructing a Confidence Interval In Exercises 17–20, you are given the sample mean and the sample standard deviation. Assume the population is normally distributed and use the t-distribution to find the margin of error and construct a 95% confidence interval for the population mean. Interpret the results.

Commute Time In a random sample of eight people, the mean commute time to work was 35.5 minutes and the standard deviation was 7.2 minute

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Textbook Question

You research prices of cell phones and find that the population mean is $431.61. In Exercise 19, does the t-value fall between -t0.95 and t0.95?

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Textbook Question

In Exercise 28, the population mean weekly time spent on homework by students is 7.8 hours. Does the t-value fall between -t0.99 and t0.99?

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Textbook Question

In Exercises 55–60, find the indicated probabilities and interpret the results.

The mean annual salary for physical therapists in the United States is about $87,000. A random sample of 50 physical therapists is selected. What is the probability that the mean annual salary of the sample is (b) more than $85,000? Assume sigma = $10,500.

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Textbook Question

In Exercises 15 and 16, find the t-value for the given values of xbar, μ, s and n.

xbar = 70.3, μ = 64.8, s = 7.1, n = 16

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Textbook Question

In Exercises 13 and 14, use the confidence interval to find the margin of error and the sample mean.

(14.7, 22.1)

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Textbook Question

In Exercises 9–12, construct the indicated confidence interval for the population mean μ using the t-distribution. Assume the population is normally distributed.

c = 0.99, xbar = 24.7, s = 4.6, n = 50

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