Ch. 7 - Hypothesis Testing with One Sample

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo

Larson 8th Edition

Ch. 7 - Hypothesis Testing with One Sample

Problema 7.2.31

Capitolo 7, Problema 7.2.31

In Exercises 29–32, test the claim about the population mean at the level of significance α. Assume the population is normally distributed.

Claim: ; μ ≠ 5880; α = 0.03; α = 413
Sample statistics: x_bar = 5771, n = 67

Guida verificata passo dopo passo

Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis is H₀: μ = 5880, and the alternative hypothesis is Hₐ: μ ≠ 5880. This is a two-tailed test because the claim specifies 'not equal to' (≠).

Step 2: Calculate the test statistic using the formula for a t-test: t = (x̄ - μ) / (s / √n), where x̄ is the sample mean, μ is the population mean under the null hypothesis, s is the sample standard deviation, and n is the sample size. Substitute the given values: x̄ = 5771, μ = 5880, s = 413, and n = 67.

Step 3: Determine the degrees of freedom (df) for the t-distribution. For a single-sample t-test, df = n - 1. In this case, df = 67 - 1 = 66.

Step 4: Find the critical t-value(s) for a two-tailed test at the significance level α = 0.03 and df = 66. Use a t-distribution table or statistical software to find the critical values. These values will define the rejection region for the null hypothesis.

Step 5: Compare the calculated t-value from Step 2 to the critical t-values from Step 4. If the calculated t-value falls in the rejection region (i.e., beyond the critical values), reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the claim.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample statistics to determine whether to reject H0 in favor of H1. In this case, the null hypothesis would state that the population mean μ equals 5880, while the alternative hypothesis claims that μ does not equal 5880.

Video consigliato:

Percorso guidato

06:21

Step 1: Write Hypotheses

Level of Significance (α)

The level of significance, denoted as α, is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. In this scenario, α is set at 0.03, indicating a 3% risk of concluding that a difference exists when there is none. This threshold helps determine the critical value for the test statistic, which is essential for making a decision regarding the null hypothesis.

Video consigliato:

Percorso guidato

04:46

Step 4: State Conclusion Example 4

Sample Mean and Standard Error

The sample mean (x̄) is the average value of a sample, which in this case is 5771. The standard error (SE) measures the dispersion of the sample mean from the population mean and is calculated using the sample standard deviation divided by the square root of the sample size (n). Understanding the sample mean and standard error is crucial for calculating the test statistic and determining whether the sample provides enough evidence to reject the null hypothesis.

Video consigliato:

Percorso guidato

05:11

Sampling Distribution of Sample Proportion

Pratica correlata

Domanda del libro di testo

In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.

Claim: p > 0.70, α=0.04. Sample statistics: p_hat = 0.64, n=225

104

views

Domanda del libro di testo

Hypothesis Testing Using Rejection Regions In Exercises 23–30, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic X^2, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the population is normally distributed.

Salaries The annual salaries (in dollars) of 15 randomly chosen senior level graphic design specialists are shown in the table at the left. At α=0.05, is there enough evidence to support the claim that the standard deviation of the annual salaries is different from \$13,056?

109

views

Domanda del libro di testo

In Exercises 29–32, test the claim about the population mean at the level of significance α. Assume the population is normally distributed.

Claim: ; μ ≤ 22,500; α = 0.01; α = 1200

Sample statistics: x_bar = 23,500, n = 45

103

views

Domanda del libro di testo

In Exercises 15–22, test the claim about the population variance or standard deviation at the level of significance Assume the population is normally distributed.

Claim: σ<40, α=0.01 . Sample statistics: s=40.8, n=12

views

Domanda del libro di testo

Explain how to test a population variance or a population standard deviation.

123

views

Domanda del libro di testo

In Exercises 7–12, find the critical value(s) and rejection region(s) for the type of chi-square test with sample size n and level of significance α.

Left-tailed test, n=24,α=0.05

views