Problem 30 - Basic Statistics for Business and Economics with Student

Question

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 oneweek 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?

Step-by-Step Solution

Answer

No, there is no significant difference at the 0.01 level.

Step 1: State the Hypotheses

First, let's establish the null and alternative hypotheses. The null hypothesis \( H_0 \) is that there is no difference in the proportions of successful connections between HTC and Mountain Communications. Mathematically, it is expressed as \( p_1 = p_2 \). The alternative hypothesis \( H_1 \) is that there is a difference in these proportions, expressed as \( p_1 eq p_2 \).

Step 2: Calculate Sample Proportions

Calculate the sample proportions for both providers. The proportion for HTC is \( \hat{p}_1 = \frac{450}{500} = 0.9 \) and for Mountain Communications is \( \hat{p}_2 = \frac{352}{400} = 0.88 \).

Step 3: Determine the Pooled Proportion

The pooled proportion \( \hat{p} \) combines the successes and total trials from both samples: \( \hat{p} = \frac{450 + 352}{500 + 400} = \frac{802}{900} \approx 0.8911 \).

Step 4: Calculate the Standard Error

The standard error (SE) of the difference between the two sample proportions is calculated by:\[SE = \sqrt{\hat{p}(1-\hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} = \sqrt{0.8911 \times (1 - 0.8911) \left( \frac{1}{500} + \frac{1}{400} \right)} \approx 0.0209\]

Step 5: Compute the Z-Statistic

The Z-statistic measures how many standard errors the difference in sample proportions is away from the hypothesized difference of 0:\[Z = \frac{\hat{p}_1 - \hat{p}_2}{SE} = \frac{0.9 - 0.88}{0.0209} \approx 0.957\]

Step 6: Determine the Critical Value and Decision

With a significance level of 0.01 for a two-tailed test, the critical Z-values are approximately ±2.576. Since the computed Z (0.957) is within this range, we fail to reject the null hypothesis.

Key Concepts

Sample Proportions

When dealing with data from a sample, one useful measure is the sample proportion. In this context, it represents the fraction of successful internet connections in a sample of trials. For HTC, the sample proportion is calculated by dividing the number of successful connections (450) by the total number of trials (500). This gives us a sample proportion of 0.9. Similarly, for Mountain Communications, the sample proportion is 0.88, calculated from 352 successful connections out of 400 trials.
The sample proportion provides a quick snapshot of success rates for each provider in this study. It helps in estimating the probability that a customer will successfully connect to the internet with each provider. It's important because we are trying to understand if there are significant differences between the providers in terms of successful connection rates.

Null and Alternative Hypotheses

The hypothesis testing process often begins with the formulation of the null and alternative hypotheses. Here, the null hypothesis (\( H_0 \)) states that there is no difference in success rates between HTC and Mountain Communications, represented mathematically as \( p_1 = p_2 \). On the other hand, the alternative hypothesis (\( H_1 \)) proposes that there is a difference in success rates, shown as \( p_1 eq p_2 \).
These hypotheses set the stage for subsequent statistical analysis. By attempting to disprove the null hypothesis, we use sample data to infer whether there's enough evidence to suggest a significant difference in internet access success rates between the two companies. Ensuring clear hypotheses is crucial because they guide the statistical test selection and help interpret results accurately.

Significance Level

The significance level in hypothesis testing signifies the probability of rejecting the null hypothesis when it is, in fact, true. In this scenario, a significance level of 0.01 indicates extreme caution, as it sets a stringent criterion of demonstrating a significant difference between the success rates. A lower significance level means we need very convincing evidence to reject the null hypothesis.
When conducting the test, we compare the calculated value (like the Z-statistic) to the critical value associated with our chosen significance level. If the calculated value falls beyond the critical thresholds for a two-tailed test, we reject the null hypothesis. A 0.01 significance level corresponds to critical Z-values of ±2.576 for this test.

Standard Error

In this statistical analysis, the standard error helps us measure the accuracy of the sample proportion difference as an estimate of the population proportion difference. It shows how much the sample proportions from the two providers might vary compared to each other.

• The calculation involves using the pooled proportion, which is the overall proportion of success from both samples combined.
• For this exercise, the pooled proportion (\( \hat{p} \)) is approximately 0.8911.

The standard error of the difference between the two sample proportions is found with the formula: \[ SE = \sqrt{\hat{p}(1-\hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \approx 0.0209 \]With this standard error, we can then compute the Z-statistic, which gauges how far our observed sample difference deviates from the assumed difference under the null hypothesis. Understanding the standard error is vital for assessing the reliability and significance of the statistical test's outcome.