9. Hypothesis Testing for One Sample

Performing Hypothesis Tests: Means

9. Hypothesis Testing for One Sample

Performing Hypothesis Tests: Means

3:09 minutes

Problem 8.3.5

Textbook Question

Finding P-values
In Exercises 5–8, either use technology to find the P-value or use Table A-3 to find a range of values for the P-value. Based on the result, what is the final conclusion?

Weights of Quarters The claim is that weights (grams) of quarters made after 1964 have a mean equal to 5.670 g as required by mint specifications. The sample size is and the test statistic is t = -3.135.

Verified step by step guidance

Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ = 5.670 g, which states that the mean weight of quarters is equal to 5.670 g. The alternative hypothesis is H₁: μ ≠ 5.670 g, which indicates a two-tailed test.

Step 2: Determine the degrees of freedom (df) for the t-distribution. The degrees of freedom are calculated as df = n - 1, where n is the sample size. Ensure you know the sample size to compute df.

Step 3: Use the given test statistic t = -3.135 and the degrees of freedom to find the P-value. You can either use statistical software or a t-distribution table (Table A-3). For a two-tailed test, double the area in the tail corresponding to |t| = 3.135.

Step 4: Compare the P-value to the significance level (α). If the P-value is less than α (commonly 0.05), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

Step 5: Based on the comparison, draw a conclusion. If the null hypothesis is rejected, conclude that there is sufficient evidence to support the claim that the mean weight of quarters is not equal to 5.670 g. If the null hypothesis is not rejected, conclude that there is insufficient evidence to refute the claim.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

P-value

The P-value is a statistical measure that helps determine the significance of results from a hypothesis test. It represents the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is true. A smaller P-value indicates stronger evidence against the null hypothesis, often leading to its rejection if it falls below a predetermined significance level, such as 0.05.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a default position, and the alternative hypothesis (H1), which represents what we aim to support. The test statistic, such as the t-value in this case, is calculated to assess the evidence against the null hypothesis.

Test Statistic

A test statistic is a standardized value that is calculated from sample data during a hypothesis test. It quantifies the difference between the observed sample statistic and the hypothesized population parameter, scaled by the standard error. In this scenario, the t-statistic of -3.135 indicates how many standard errors the sample mean is away from the hypothesized mean, providing a basis for determining the P-value.

Watch next

Master Standard Deviation (σ) Known with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

A travel analyst says that the mean price of a meal for a family of 4 in a resort restaurant is at most $100. A random sample of 33 meal prices for families of 4 has a mean of $110 and a standard deviation of $19. At α=0.01, is there enough evidence to reject the analyst’s claim?

views

Textbook Question

[APPLET] A researcher claims that the mean age of the residents of a small town is more than 38 years. The ages (in years) of a random sample of 30 residents are listed below. At α=0.10, is there enough evidence to support the researcher’s claim? Assume the population standard deviation is 9 years.

views

Textbook Question

Hypothesis Testing. In Exercises 17–19, apply the central limit theorem to test the given claim. (Hint: See Example 3.)

Adult Sleep Times (hours) of sleep for randomly selected adult subjects included in the National Health and Nutrition Examination Study are listed below. Here are the statistics for this sample: n = 12, x_bar = 6.8 hours, s = 20 hours. The times appear to be from a normally distributed population. A common recommendation is that adults should sleep between 7 hours and 9 hours each night. Assuming that the mean sleep time is 7 hours, find the probability of getting a sample of 12 adults with a mean of 6.8 hours or less. What does the result suggest about a claim that “the mean sleep time is less than 7 hours”?

4 8 4 4 8 6 9 7 7 10 7 8

views

Textbook Question

Test Statistic and Critical Value The statistics for the sample data in Exercise 1 are n = 15, x_bar = 6.133333, and s = 8.862978, where the units are millions of dollars. Find the test statistic and critical value(s) for a test of the claim that the salaries are from a population with a mean greater than 5 million dollars. Assume that a 0.05 significance level is used.

views

Textbook Question

Technology

In Exercises 9–12, test the given claim by using the display provided from technology. Use a 0.05 significance level. Identify the null and alternative hypotheses, test statistic, P-value (or range of P-values), or critical value(s), and state the final conclusion that addresses the original claim.

Peanut Butter Cups Data Set 38 “Candies” includes weights of Reese’s peanut butter cups. The accompanying Statdisk display results from using all 38 weights to test the claim that the sample is from a population with a mean equal to 8.953 g.

views

Textbook Question

Technology

Tower of Terror Data Set 33 “Disney World Wait Times” includes wait times (minutes) for the Tower of Terror ride at 5:00 PM. Using the first 40 times to test the claim that the mean of all such wait times is more than 30 minutes, the accompanying Excel display is obtained.

views

Textbook Question

Testing Hypotheses

In Exercises 13–24, assume that a simple random sample has been selected and test the given claim. Unless specified by your instructor, use either the P-value method or the critical value method for testing hypotheses. Identify the null and alternative hypotheses, test statistic, P-value (or range of P-values), or critical value(s), and state the final conclusion that addresses the original claim.

Systolic Blood Pressure Systolic blood pressure levels above 120 mm Hg are considered to be high. For the 300 systolic blood pressure levels listed in Data Set 1 “Body Data” from Appendix B, the mean is 122.96000 mm Hg and the standard deviation is 15.85169 mm Hg. Use a 0.01 significance level to test the claim that the sample is from a population with a mean greater than 120 mm Hg.

views

Textbook Question

Testing Hypotheses

Diastolic Blood Pressure Diastolic blood pressure levels of 60 mm Hg or lower are considered to be too low. For the 300 diastolic blood pressure levels listed in Data Set 1 “Body Data” from Appendix B, the mean is 70.75333 mm Hg and the standard deviation is 11.61618 mm Hg. Use a 0.01 significance level to test the claim that the sample is from a population with a mean greater than 60 mm Hg.

views

Show more practices