In contract negotiations a company claims that a new...

In contract negotiations a company claims that a new incentive scheme has resulted in average weekly earnings of at least $$\$ 400$$ for all customer service workers. A union representative takes a random sample of 15 workers and finds that their weekly earnings have an average of $$\$ 381.35$$ and a standard deviation of $$\$ 48.60$$. Assume a normal distribution. a. Test the company's claim. b. If the same sample results had been obtained from a random sample of 50 employees, could the company's claim be rejected at a lower significance level than that used in part a?

In contract negotiations a company claims that a new incentive scheme has resulted in average weekly earnings of at least $$\$ 400$$ for all customer service workers. A union representative takes a random sample of 15 workers and finds that their weekly earnings have an average of $$\$ 381.35$$ and a standard deviation of $$\$ 48.60$$. Assume a normal distribution.
a. Test the company's claim.
b. If the same sample results had been obtained from a random sample of 50 employees, could the company's claim be rejected at a lower significance level than that used in part a?

Key Concepts

Effect of Sample Size on Hypothesis Testing

Sample size plays a critical role in hypothesis testing by influencing the standard error of the estimated parameter. Larger sample sizes lead to smaller standard errors, yielding more precise estimates and potentially stronger conclusions in hypothesis testing. A larger sample size, as in the second scenario with 50 employees, would provide more information and might allow for the company's claim to be rejected even at a lower significance level.

Significance Level

The significance level is the threshold used to determine whether to reject the null hypothesis. It represents the probability of rejecting the null hypothesis when it is actually true (Type I error). In hypothesis testing, if the computed p-value is less than the significance level, the null hypothesis is rejected, indicating statistically significant evidence against it.

t-Distribution

The t-distribution is a probability distribution that is used instead of the normal distribution when the sample size is small and the population standard deviation is unknown. It accounts for additional uncertainty due to estimation of the standard error and is defined by degrees of freedom, which depend on the sample size minus one.

Null and Alternative Hypotheses

This key concept refers to the formulation of statements to be tested in a statistical test. The null hypothesis (H?) represents the claim that there is no effect or that a parameter is at least or equal to a certain value, while the alternative hypothesis (H?) represents the claim that contradicts the null. In the context of the problem, the null hypothesis would assert that the average weekly earnings are at least $400 and the alternative would suggest that they are less than $400.

One?Sample t-Test

A one-sample t-test is used when comparing the sample mean to a known or hypothesized population mean, especially when the population standard deviation is unknown and the sample size is small. This test employs the t-distribution to determine how likely it is that the sample mean would occur if the null hypothesis were true.

Transcript

Please give Ace some feedback