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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3:42 minutes

Problem 6.2.30

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?

Verified step by step guidance

Step 1: Identify the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis typically states that the sample mean is equal to the population mean (H0: μ = 7.8), while the alternative hypothesis suggests a difference (H1: μ ≠ 7.8).

Step 2: Calculate the t-value using the formula:

t = \frac{x̄ - μ}{\frac{s}{\sqrt{n}}}

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

Step 3: Determine the critical t-values for a 99% confidence level (two-tailed test). These values are denoted as -t_0.99 and t_0.99. Use a t-distribution table or statistical software to find these values based on the degrees of freedom (df = n - 1).

Step 4: Compare the calculated t-value from Step 2 to the critical t-values from Step 3. If the t-value falls between -t_0.99 and t_0.99, the null hypothesis is not rejected. Otherwise, the null hypothesis is rejected.

Step 5: Conclude whether the t-value falls within the range of -t_0.99 and t_0.99, and interpret the result in the context of the problem. This will help determine if the sample mean significantly differs from the population mean.

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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 weekly time spent on homework by all students, which is given as 7.8 hours. Understanding the population mean is crucial for making inferences about the data and comparing sample statistics.

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 statistically different from each other. In this case, the t-value will help assess whether the observed data significantly deviates from the population mean.

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 and the population standard deviation is unknown. The critical values, such as -t0.99 and t0.99, define the cutoff points for determining statistical significance in hypothesis testing.

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Related Practice

Textbook Question

Use technology to find the standard deviation of the set of 36 sample means. How does it compare with the standard deviation of the ages found in Exercise 5? Does this agree with the result predicted by the Central Limit Theorem?

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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 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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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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