There are two major Internet providers in the Colorado...

There are two major Internet providers in the Colorado Springs, Colorado, area, one called HTC and the other Mountain Communications. We want to investigate whether there is a difference in the proportion of times a customer is able to access the Internet. During a oneweek period, 500 calls were placed at random times throughout the day and night to HTC. A connection was made to the Internet on 450 occasions. A similar one-week study with Mountain Communications showed the Internet to be available on 352 of 400 trials. At the .01 significance level, is there a difference in the percent of time that access to the Internet is successful?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to assess evidence provided by sample data in order to make inferences about a population parameter. It involves formulating a null hypothesis, which represents a position of no effect or no difference, and an alternative hypothesis representing a different claim. The process typically includes calculating a test statistic and comparing it to a critical value or using a p-value to determine the likelihood of the observed result under the null hypothesis.

Null Hypothesis

The null hypothesis is a foundational concept in hypothesis testing. It is a default statement that there is no effect or no difference between populations or groups. In the context of comparing two proportions, the null hypothesis posits that the two population proportions are equal, setting the stage for determining whether any observed difference is statistically significant.

Alternative Hypothesis

The alternative hypothesis is the statement that contradicts the null hypothesis. It suggests that there is an effect or a difference between populations. In cases comparing two proportions, the alternative hypothesis typically asserts that the proportions are not equal, and it guides the direction of the test (e.g., two-tailed for any difference).

Two-proportion z-test

A two-proportion z-test is a statistical method for comparing the differences between two independent sample proportions. This test assesses whether the observed difference between the two proportions is large enough to be considered statistically significant, taking into account the variability and sample sizes. It is particularly useful when comparing rates or proportions between two groups.

Significance Level

The significance level, often denoted as alpha (?), is the threshold set by the researcher to determine when to reject the null hypothesis. It represents the acceptable probability of committing a Type I error, which is the error of rejecting a true null hypothesis. Common significance levels include 0.05 and 0.01, with lower levels requiring stronger evidence to reject the null hypothesis.

P-value

The p-value is the probability of obtaining test results at least as extreme as the ones observed, assuming that the null hypothesis is true. It serves as a measure of the strength of the evidence against the null hypothesis. When the p-value is less than the pre-determined significance level, the null hypothesis is rejected, indicating that the observed effect is statistically significant.

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