For the United States, the mean monthly Internet bill is ...

For the United States, the mean monthly Internet bill is $$\$ 32.79$$ per household (CNBC, January 18,2006 ). A sample of 50 households in a southern state showed a sample mean of $$\$ 30.63$$. Use a population standard deviation of $$\sigma=\$ 5.60$$. a. Formulate hypocheses for a test to determine whether the sample data support the conclusion that the mean monthly Intemet bill in the southern state is less than the national mean of $$\$ 32.79$$. b. What is the value of the test statistic? c. What is the p-value? d. At $\alpha=.01$, what is your conclusion?

For the United States, the mean monthly Internet bill is $$\$ 32.79$$ per household (CNBC, January 18,2006 ). A sample of 50 households in a southern state showed a sample mean of $$\$ 30.63$$. Use a population standard deviation of $$\sigma=\$ 5.60$$.
a. Formulate hypocheses for a test to determine whether the sample data support the conclusion that the mean monthly Intemet bill in the southern state is less than the national mean of $$\$ 32.79$$.
b. What is the value of the test statistic?
c. What is the p-value?
d. At $\alpha=.01$, what is your conclusion?

Key Concepts

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) represents the default statement to be tested, typically stating that there is no effect or no difference. Conversely, the alternative hypothesis (H? or Ha) is what the researcher wants to provide evidence for. In problems where a specific direction is suspected (e.g., the state’s mean is less than the national mean), a one?tailed test is used. Clearly formulating these hypotheses is critical for determining the rejection region and interpreting the results.

One-Sample z-Test

A one-sample z-test is used when comparing a sample mean to a known population mean, particularly when the population standard deviation is known or the sample size is large. This test relies on the assumption that the sampling distribution of the sample mean is normal, thanks to the Central Limit Theorem, and it makes it possible to compute the z statistic to measure how far the sample mean deviates from the hypothesized population mean in standard error units.

Test Statistic (z-value)

The test statistic in hypothesis testing quantifies the difference between the sample statistic and the hypothesized population parameter, scaled by the standard error. In a z-test, this statistic follows a standard normal distribution under the null hypothesis. Calculating the z-value allows you to measure how many standard errors away from the hypothesized mean the sample mean lies.

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. It quantifies the evidence against the null hypothesis. A smaller p-value indicates stronger evidence for rejecting the null hypothesis.

Significance Level and Decision Rule

The significance level, denoted by alpha (?), is the probability threshold set for rejecting the null hypothesis. It is the maximum risk of a Type I error the researcher is willing to accept. Based on this threshold, the p-value is compared to ?: if the p-value is less than or equal to ?, the null hypothesis is rejected; otherwise, it is not rejected. This decision rule is essential for deriving the final conclusion from the test outcome.

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