9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

2:52 minutes

Problem 8.4.3

Textbook Question

Minting Dollar Coins For the sample data from Exercise 1, we get a P-value of 0.0041 when testing the claim that σ < 0.04000 g.

What should we conclude about the null hypothesis?
What should we conclude about the original claim?
What do these results suggest about the new minting process?

Verified step by step guidance

Step 1: Understand the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis (H₀) states that σ ≥ 0.04000 g, while the alternative hypothesis (H₁) states that σ < 0.04000 g. This is a one-tailed test for the population standard deviation.

Step 2: Interpret the P-value. The P-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample data, assuming the null hypothesis is true. Here, the P-value is 0.0041.

Step 3: Compare the P-value to the significance level (α). If the P-value is less than the chosen significance level (commonly α = 0.05), we reject the null hypothesis. In this case, 0.0041 < 0.05, so we reject H₀.

Step 4: Conclude about the original claim. Since the null hypothesis is rejected, there is sufficient evidence to support the claim that σ < 0.04000 g. This suggests that the variability in the minting process is less than 0.04000 g.

Step 5: Interpret the results in context. The results suggest that the new minting process is more consistent (less variable) than the threshold of 0.04000 g. This could indicate an improvement in the precision of the minting process.

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Key Concepts

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

Null Hypothesis

The null hypothesis is a statement that there is no effect or no difference, and it serves as the default assumption in statistical testing. In this context, it posits that the standard deviation (σ) of the minting process is greater than or equal to 0.04000 g. A low P-value indicates strong evidence against the null hypothesis, suggesting it may be rejected in favor of the alternative hypothesis.

P-value

The P-value is a statistical measure that helps determine the significance of results in hypothesis testing. It represents the probability of observing the data, or something more extreme, assuming the null hypothesis is true. A P-value of 0.0041 indicates a very low probability of obtaining such results if the null hypothesis were correct, leading to the conclusion that the null hypothesis is likely not true.

Alternative Hypothesis

The alternative hypothesis is the statement that contradicts the null hypothesis, suggesting that there is an effect or a difference. In this scenario, it would claim that the standard deviation (σ) is less than 0.04000 g. The results of the test, particularly the low P-value, provide evidence supporting the alternative hypothesis, indicating that the new minting process may be more precise than previously thought.

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Related Practice

Textbook Question

Finding P-Values

In Exercises 13–16, do the following:

i. Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed.

ii. Find the P-value. (See Figure 8-3.)

iii. Using a significance level of α = 0.05 should we reject H0 or should we fail to reject H0?

The test statistic of z = -1.60 is obtained when testing the claim that p ≠ 0.455.

Textbook Question

Finding Critical Values

In Exercises 17–20, refer to the information in the given exercise and use a 0.05 significance level for the following.

a. Find the critical value(s).

b. Should we reject H0 or should we fail to reject H0?

Exercise 14

Textbook Question

Finding Critical Values

In Exercises 17–20, refer to the information in the given exercise and use a 0.05 significance level for the following.

a. Find the critical value(s).

b. Should we reject H0 or should we fail to reject H0?

Exercise 15

Textbook Question

Finding Critical Values

In Exercises 17–20, refer to the information in the given exercise and use a 0.05 significance level for the following.

a. Find the critical value(s).

b. Should we reject H0 or should we fail to reject H0?

Exercise 16

Textbook Question

Finding Critical Values of (chi)^2 For large numbers of degrees of freedom, we can approximate critical values of as follows:

(chi)^2 = (1/2)(z + sqrt(2k-1))

Here k is the number of degrees of freedom and z is the critical value(s) found from technology or Table A-2. In Exercise 12 “Spoken Words” we have df = 55, so Table A-4 does not list an exact critical value. If we want to approximate a critical value of (chi)^2 in the right-tailed hypothesis test with α = 0.01 and a sample size of 56, we let k =55 with z = 2.33 (or the more accurate value of z = 2.326348 found from technology). Use this approximation to estimate the critical value of for Exercise 12. How close is it to the critical value of (chi)^2 = 82.292 obtained by using Statdisk and Minitab?

Textbook Question

RESAMPLING

c. When testing a claim about a proportion or mean or standard deviation, what is an important advantage of using a resampling method instead of the parametric method described in the preceding sections of this chapter?

Textbook Question

At Least As Extreme A random sample of 860 births in New York State included 426 boys, and that sample is to be used for a test of the common belief that the proportion of male births in the population is equal to 0.512.

a. In testing the common belief that the proportion of male babies is equal to 0.512, identify the values of p^ and p.

Textbook Question

In Exercises 13–16, write a statement that interprets the P-value and includes a conclusion about linear correlation.

Using the data from Exercise 6 “Airport Data Speeds,” the P-value is 0.003.

Show more practices

Minting Dollar Coins For the sample data from Exercise 1, we get a P-value of 0.0041 when testing the claim that σ < 0.04000 g.What should we conclude about the null hypothesis?What should we conclude about the original claim?What do these results suggest about the new minting process?

Key Concepts

Null Hypothesis

P-value

Alternative Hypothesis

Watch next

Minting Dollar Coins For the sample data from Exercise 1, we get a P-value of 0.0041 when testing the claim that σ < 0.04000 g.

What should we conclude about the null hypothesis?
What should we conclude about the original claim?
What do these results suggest about the new minting process?