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In a survey of mobile phone owners, \(53 \%\) of iPhone users and \(42 \%\) of Android phone users indicated that they upgraded their phones at least every two years ("Americans Split on How Often They Upgrade Their Smartphones," July 8, \(2015,\) www.gallup.com, retrieved December 15,2016 ). The reported percentages were based on large samples that were thought to be representative of the population of iPhone users and the population of Android phone users. The sample sizes were 8234 for the iPhone sample and 6072 for the Android phone sample. Suppose you want to decide if there is evidence that the proportion of iPhone owners who upgrade their phones at least every two years is greater than this proportion for Android phone users. (Hint: See Example 11.5.) a. What hypotheses should be tested to answer the question of interest? b. Are the two samples large enough for the large-sample test for a difference in population proportions to be appropriate? Explain. c. Based on the following Minitab output, what is the value of the test statistic and what is the value of the associated \(P\) -value? If a significance level of 0.01 is selected for the test, will you reject or fail to reject the null hypothesis? d. Interpret the result of the hypothesis test in the context of this problem.

Step by step solution

01

a. Hypotheses Formulation

Let \(p_1\) be the proportion of iPhone users that upgrade their phones at least every two years, and \(p_2\) be the proportion of Android users that upgrade their phones at least every two years. We want to test if there is evidence that the proportion of iPhone owners who upgrade their phones at least every two years is greater than this proportion for Android phone users. In other words, we want to test if \(p_1 > p_2\). The null hypothesis (\(H_0\)) states that there is no difference between the proportions, and the alternative hypothesis (\(H_a\)) states that there is a difference. \(H_0 : p_1 - p_2 = 0\) \(H_a : p_1 - p_2 > 0\)

02

b. Assessing Sample Size Appropriateness for the Large-Sample Test

The large-sample test for a difference in population proportions is appropriate when the sample sizes for both populations are large enough. This is typically satisfied when the following four conditions are met: 1. \(n_1p_1 \geq 5\) 2. \(n_1(1-p_1) \geq 5\) 3. \(n_2p_2 \geq 5\) 4. \(n_2(1-p_2) \geq 5\) Using the given sample sizes and proportions: 1. \(8234 \times 0.53 \geq 5\), which is true. 2. \(8234 \times (1-0.53) \geq 5\), which is true. 3. \(6072 \times 0.42 \geq 5\), which is true. 4. \(6072 \times (1-0.42) \geq 5\), which is true. Since all four conditions are met, the sample sizes are large enough for the large-sample test for a difference in population proportions to be appropriate.

03

c. Calculating Test Statistic, P-value, and Making a Decision

Since we are not given the Minitab output, we will compute it manually: The test statistic is given by: \(z = \frac{\bar{p}_1 - \bar{p}_2}{\sqrt{\frac{\bar{p}_1(1-\bar{p}_1)}{n_1}+\frac{\bar{p}_2(1-\bar{p}_2)}{n_2}}}\) where \(\bar{p}_1 = 0.53\), \(\bar{p}_2 = 0.42\), \(n_1 = 8234\) and \(n_2 = 6072\). Plugging in the values, we get: \(z = \frac{0.53 - 0.42}{\sqrt{\frac{0.53\times (1-0.53)}{8234}+\frac{0.42\times (1-0.42)}{6072}}}\) \(z \approx 10.18\) Now, we need to find the P-value to make a decision. Since we have a one-tailed test, we use the standard normal distribution table to find the P(z > 10.18). In our case, this value is virtually 0 (because 10.18 is extremely large for standard normal distribution). Since the P-value (approximately 0) is less than the significance level (0.01), we reject the null hypothesis.

04

d. Interpreting the Result

By rejecting the null hypothesis, we can conclude that there is significant evidence at the 0.01 significance level that the proportion of iPhone users who upgrade their phones at least every two years is greater than the proportion of Android users who upgrade their phones at least every two years.

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Key Concepts

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

Population Proportion

When we refer to the population proportion, we are talking about the fraction or percentage of the population that showcase a certain attribute or behavior. In our context, population proportion pertains to the percentage of iPhone or Android users who choose to upgrade their phones at least every two years.

To give you a clearer idea, the population proportion is usually denoted by the symbol 'p.' When we conduct studies or surveys, we can't ask every single individual in the population, so we take a sample and calculate the sample proportion, depicted as \( \bar{p} \). If the sample is representative, this sample proportion gives us a good estimate of the actual population proportion. It’s critical to ensure that our sample is large and unbiased, so conclusions drawn from the sample are likely to be an accurate reflection of the entire population.

Large-Sample Test

A large-sample test is a type of hypothesis test applied when the sample size is sufficiently large to use normal approximation. This test is particularly relevant when dealing with population proportions, as it allows us to make inferences about the population based on our sample data.

The 'large enough' criteria for sample sizes are guided by rules of thumb that expect the product of the sample size and the population proportion (both the success and failure—think 'p' and '1-p') to be greater than 5. When these conditions are met, we can confidently use the Z-test for comparing population proportions. In the context of our exercise, we verified that both iPhone and Android user samples were large enough to proceed with a large-sample test, thus providing reliable results for the proportion comparison.

Statistical Significance

Now, let’s talk about statistical significance. This concept helps us to determine whether the findings of our sample study can be generalized to the larger population. To assess statistical significance, we compare the P-value of our test statistic to a predetermined level of significance, often denoted by alpha (\(\alpha\)).

If our P-value is less than or equal to \(\alpha\), we reject the null hypothesis, suggesting that our findings are unlikely to be due to random chance and should be considered statistically significant. In our mobile phone upgrade study, the P-value was so minuscule that it virtually hit zero—far below the \(\alpha=0.01\), indicating a statistically significant difference between the proportions of iPhone and Android users upgrading their phones. Hence, we can confidently say that iPhone users indeed upgrade their phones at a higher rate than Android users.