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The article "How to Block Nuisance Calls" (The Guardian, November 7,2015 ) reported that in a survey of mobile phone users, \(70 \%\) of those surveyed said they had received at least one nuisance call to their mobile phone in the last month. Suppose that this estimate was based on a representative sample of 600 mobile phone users. These data can be used to determine if there is evidence that more than two-thirds of all mobile phone users have received at least one nuisance call in the last month. The large-sample test for a population proportion was used to test \(H_{0}: p=0.667\) versus \(H_{i}: p>\) 0.667 . The resulting \(P\) -value was \(0.043 .\) Using a significance level of \(0.05,\) the null hypothesis was rejected. a. Based on the hypothesis test, what can you conclude about the proportion of mobile phone users who received at least one nuisance call on their mobile phones within the last month? 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

a. Determine the Conclusion

Since the \(P\)-value of the test (0.043) is less than the given significance level (\(\alpha = 0.05\)), we reject the null hypothesis. Hence, we can conclude that there is enough evidence to suggest that more than two-thirds (\(>\) 0.667) of all mobile phone users have received at least one nuisance call within the last month.

02

b. Support for the Alternative Hypothesis

The alternative hypothesis states that the true proportion of mobile phone users who received a nuisance call in the past month is greater than 0.667. Our result supports the alternative hypothesis, but the P-value (0.043) is close to the significance level, which means that the data provides only moderate support for the alternative hypothesis.

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c. Evidence Against the Null Hypothesis

Since the P-value is less than the significance level, the data provides evidence against the null hypothesis. Nonetheless, the data provides only moderate evidence against the null hypothesis instead of strong evidence, as the P-value is close to the significance level. In conclusion: a. We can conclude that there is enough evidence to support that more than two-thirds of all mobile phone users have received at least one nuisance call within the last month. b. The data provide only moderate support for the alternative hypothesis. c. The data provide moderate evidence against the null hypothesis.

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Key Concepts

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

Population Proportion

In statistics, the population proportion refers to the fraction of items in a population that possess a particular attribute or characteristic. It's denoted as p and can range between 0 and 1 (or 0% and 100%). For instance, in the survey regarding nuisance calls, the estimated population proportion of mobile phone users who received such a call in the past month is 0.7 (or 70%). To make valid conclusions about the population, this estimate is usually derived from a representative sample.

When we talk about hypotheses testing involving population proportions, we assess whether the sample proportion reflects a real effect in the population or if it occurred by chance. In our exercise, the hypothesis test focused on determining if the true population proportion of nuisance calls was greater than the critical value of two-thirds (or approximately 0.667).

P-value

The P-value is a critical concept in hypothesis testing that quantifies the probability of obtaining test results at least as extreme as the results observed during the study, assuming that the null hypothesis is true. It's a value between 0 and 1 and helps decide whether to reject the null hypothesis.

In simpler terms, a small P-value, typically less than the chosen significance level (like our example's 0.043 < 0.05), suggests that the observed data is unlikely under the null hypothesis. As a result, we gain confidence in favoring the alternative hypothesis. Interpretation of the P-value requires caution: a P-value less than the significance level doesn't prove the alternative hypothesis; it only indicates that the data isn't consistent with the null hypothesis.

Significance Level

The significance level, commonly symbolized as α, is a threshold that determines the likelihood of rejecting a true null hypothesis. It's a predetermined value that the researchers set, often at 0.05 or 5%. Lowering the significance level makes the test more conservative and reduces the risk of wrongly rejecting the null hypothesis (Type I error).

During a hypothesis test, if your P-value is lower than the significance level, you have sufficient grounds to reject the null hypothesis. In the exercise at hand, the significance level was set at 0.05, and since the P-value of 0.043 was lower, it led to the null hypothesis being rejected.

Null Hypothesis

The null hypothesis (H₀) is a statement of no effect or no difference. It's a skeptical stance or a starting assumption that there is nothing unusual occurring with regards to the variable in question. In statistical analysis, it acts as a claim to be tested with an aim to either reject or fail to reject it based on the evidence at hand.

For instance, in the nuisance calls scenario, the null hypothesis (H₀: p=0.667) proposes that the true proportion of all mobile phone users who received at least one nuisance call in the last month is the same as or less than two-thirds. The analysis proceeds by examining whether the sample data supports this hypothesis or suggests an alternative.

Alternative Hypothesis

Contrasting with the null hypothesis, the alternative hypothesis (H₁ or H_a) posits that there is some statistical effect or difference. It's what researchers expect to find if the null hypothesis is invalid. In our context, the alternative hypothesis (H₁: p > 0.667) asserts that the population proportion of mobile users who received nuisance calls is greater than two-thirds of all users.

The alternative hypothesis represents the outcomes that we consider significant in the real world. It's worth noting that aligning the support for the alternative hypothesis with real-world significance requires careful consideration beyond just statistical calculations.