Faculty Classroom Hours The dean of a university estimates that the mean number of classroom hours per week for full-time faculty is 11.0. As a member of the student council, you want to test this claim. A random sample of the number of classroom hours for eight full-time faculty for one week is shown in the table at the left. At α=0.01, can you reject the dean’s claim?

검증된 단계별 안내

Step 1: Formulate the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ = 11.0, which states that the mean number of classroom hours is 11.0. The alternative hypothesis is H₁: μ ≠ 11.0, which states that the mean number of classroom hours is not 11.0.

Step 2: Calculate the sample mean (x̄) using the provided data. Add all the classroom hours from the table and divide by the number of observations (n = 8). Use the formula:

x̄ = \frac{\sum x}{n}

Step 3: Calculate the sample standard deviation (s) using the formula:

s = \sqrt{\frac{\sum (x - x̄)^{2}}{n - 1}}

. This measures the variability of the classroom hours.

Step 4: Compute the test statistic (t) using the formula:

t = \frac{x̄ - μ}{\frac{s}{\sqrt{n}}}

, where μ is the hypothesized mean (11.0), x̄ is the sample mean, s is the sample standard deviation, and n is the sample size.

Step 5: Compare the calculated t-value to the critical t-value at α = 0.01 for a two-tailed test with degrees of freedom (df = n - 1). If the absolute value of the calculated t-value exceeds the critical t-value, reject the null hypothesis; otherwise, fail to reject the null hypothesis.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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) that represents a default position, and an alternative hypothesis (H1) that represents what we aim to prove. In this case, the null hypothesis would state that the mean classroom hours are equal to 11.0, while the alternative would suggest they are not. The goal is to determine whether the sample data provides sufficient evidence to reject the null hypothesis.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether to reject the null hypothesis. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. In this scenario, α is set at 0.01, indicating a 1% risk of concluding that the mean classroom hours differ from 11.0 when they actually do not. A lower α value means a stricter criterion for evidence against the null hypothesis.

Sample Mean and Standard Deviation

The sample mean is the average of the observed values in a sample, providing an estimate of the population mean. The standard deviation measures the dispersion of the sample data around the mean. In this context, calculating the sample mean of the classroom hours will help assess whether it significantly differs from the dean's claim of 11.0 hours. The standard deviation will also be crucial for determining the variability of the sample and for conducting the hypothesis test.