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The article "Irritated by Spam? Get Ready for Spit" (USA Today, November 10,2004 ) predicts that "spit," spam that is delivered via Internet phone lines and cell phones, will be a growing problem as more people turn to web- based phone services. In a poll of 5,500 cell phone users, \(20 \%\) indicated that they had received commercial messages and ads on their cell phones. These data were used to test \(H_{o}: p=0.13\) versus \(H_{a}: p>0.13\) where 0.13 was the proportion reported for the previous year. The null hypothesis was rejected. a. Based on the hypothesis test, what can you conclude about the proportion of cell phone users who received commercial messages and ads on their cell phones in the year the poll was conducted? b. Is it reasonable to say that the data provide strong support for the alternative hypothesis? c. Is it reasonable to say that the data provide strong evidence against the null hypothesis?

Step by step solution

01

Understanding the research hypotheses

Here, we have two hypotheses given. The null hypothesis (\(H_{0}\)) is \(p=0.13\), i.e., the percentage of people receiving spam calls remains the same as the previous year (13%). The alternative hypothesis (\(H_{a}\)) is \(p>0.13\), i.e., the percentage of people receiving spam calls has increased compared to the previous year.

02

Analyzing the sample data

In the sample data of 5,500 users, 20% of them reported receiving commercial messages and ads on their cell phones. This percentage is higher than the one stated in the null hypothesis, which is 13%.

03

Conclusions from the hypothesis test

As the null hypothesis is rejected based on the results of the hypothesis test, one can conclude that the percentage of cell phone users receiving commercial messages has increased from the previous year.

04

Evaluating the strength of evidence

While it is clear that the data does not support the null hypothesis, whether it provides strong evidence for the alternative hypothesis or against the null hypothesis depends on the significance level of the test, which is not provided in this problem. Usually, a lower p-value (below 0.05 or 0.01) is considered as strong evidence against the null hypothesis and for the alternative hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

In the context of hypothesis testing, the null hypothesis, denoted as \(H_0\), represents the current scenario or baseline that you assume to be true until the data suggests otherwise. It's like a statement of no effect or no difference. In the given problem, the null hypothesis is \(p = 0.13\), which means the proportion of cell phone users receiving spam calls has not changed from the previous year’s rate of 13%. This is a key starting point in hypothesis testing because it establishes a reference point to compare the current sample data. Rejection of the null hypothesis based on statistical evidence indicates that there is a significant change or effect.

Alternative Hypothesis

The alternative hypothesis, expressed as \(H_a\), is what you suspect might be true instead of the null hypothesis. It is an assertion that indicates a possible change, difference, or effect that the test seeks to support with evidence. In this exercise, the alternative hypothesis is \(p > 0.13\). This proposes that the proportion of users receiving spam has increased from last year’s 13%. Establishing an alternative hypothesis is crucial as it gives the hypothesis test direction and purpose—specifically, you are testing if there has been an increase in the proportion of users experiencing spam based on the data collected.

Significance Level

The significance level, often denoted by \(\alpha\), is the threshold that the p-value must meet or fall below in order to reject the null hypothesis. It reflects the degree of risk you are willing to take of being wrong when rejecting the null hypothesis. Commonly, it is set at 0.05 or 0.01. Although the exercise does not provide a specific significance level, one needs to assume or choose an appropriate \(\alpha\) value. A lower significance level like 0.01 indicates stronger evidence is required to reject the null hypothesis. It ensures a safer boundary against Type I errors, which occur when a true null hypothesis is incorrectly rejected.

Proportion Testing

Proportion testing involves comparing the sample proportion to a hypothesized population proportion. This is done to determine if there is any significant difference between the two. In this case, the poll showed that 20% (0.20) of cell phone users received spam, higher than the hypothesized 13% (0.13) from the null hypothesis. To perform proportion testing, the sample data is used in conjunction with a standard test statistic, such as the Z-test, to determine the p-value. This p-value helps to assess whether the observed proportion is statistically significantly different from the hypothesized proportion under the null hypothesis.