Fowle Marketing Research, Inc., bases charges to a client on...

Fowle Marketing Research, Inc., bases charges to a client on the assumption that telephone surveys can be completed in a mean time of 15 minutes or less, If a longer mean survey time is necessary, a premium rate is charged. A sample of 35 surveys provided the survey times shown in the file named Fowle, Based upon past studies, the population standard deviation is assumed known with $\sigma=4$ minutes. Is the premium rate justified? a. Formulate the null and alternative hypotheses for this application. b. Compute the value of the test statistic. c. What is the $p$ -value? d. $\quad$ At $\alpha=.01,$ what is your conclusion?

Fowle Marketing Research, Inc., bases charges to a client on the assumption that telephone surveys can be completed in a mean time of 15 minutes or less, If a longer mean survey time is necessary, a premium rate is charged. A sample of 35 surveys provided the survey times shown in the file named Fowle, Based upon past studies, the population standard deviation is assumed known with $\sigma=4$ minutes. Is the premium rate justified?
a. Formulate the null and alternative hypotheses for this application.
b. Compute the value of the test statistic.
c. What is the $p$ -value?
d. $\quad$ At $\alpha=.01,$ what is your conclusion?

Key Concepts

Hypothesis Testing

This is a fundamental procedure used to make inferences about a population parameter based on sample data. It involves setting up a null hypothesis, which represents a baseline assumption, and an alternative hypothesis, which reflects the claim against the baseline. Decisions are then made by comparing a test statistic to a critical value or by calculating a p-value.

Null and Alternative Hypotheses

The null hypothesis (H?) typically posits that there is no effect or no difference, while the alternative hypothesis (H?) suggests the presence of an effect or a departure from the standard. In the context of testing a population mean, these hypotheses are framed to determine if the actual mean exceeds, is less than or differs from the assumed value.

One-Sample Z-Test

When the population standard deviation is known and the sample size is sufficiently large, a one-sample z-test is used to determine whether there is a statistically significant difference between the sample mean and a known population mean. This test calculates a z-score by standardizing the difference between the sample mean and the hypothesized population mean.

Test Statistic

The test statistic is a numerical summary of the data that is compared against a critical value from a statistical distribution. For a one-sample z-test, the test statistic is computed by dividing the difference between the sample mean and the hypothesized mean by the standard error (which is the population standard deviation divided by the square root of the sample size).

p-Value

The p-value represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. It is a key metric in determining statistical significance; a small p-value indicates that the observed data is unlikely under the null hypothesis.

Significance Level (?)

The significance level is a predetermined threshold for rejecting the null hypothesis. Commonly set at values like 0.01 or 0.05, it defines the probability of committing a Type I error, which is rejecting the null hypothesis when it is actually true.

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