Webcredible, a UK-based consulting firm specializ- ing in websites, intranets, mobile devices, and applications, conducted a survey of 1,132 mobile phone users between February and April 2009. The survey found that 52% of mo- bile phone users are now using the mobile Internet. (Data extracted from “Email and Social Networking Most Popular Mobile Internet Activities. The authors of the article imply that the survey proves that more than half of all mobile phone users are now using the mobile Internet.

a. Use the five-step p-value approach to hypothesis testing and a 0.05 level of significance to try to prove that more than half of all mobile phone users are now using the mo- bile Internet.
b. Based on your result in (a), is the claim implied by the authors valid?
c. Suppose the survey found that 53% of mobile phone users are now using the mobile Internet. Repeat parts (a) and (b).
d. Compare the results of (b) and (c).

a. To test whether more than half of all mobile phone users are now using the mobile Internet, we can use the five-step p-value approach to hypothesis testing.

Step 1: State the hypotheses.
Null hypothesis (H0): p ≤ 0.5 (less than or equal to half of all mobile phone users are using the mobile Internet)
Alternative hypothesis (Ha): p > 0.5 (more than half of all mobile phone users are using the mobile Internet)

Step 2: Formulate an analysis plan.
We will use a one-sample proportion test to analyze the survey data.

Step 3: Analyze sample data.
Based on the survey, 52% of the 1,132 mobile phone users are using the mobile Internet. This gives us an observed proportion of p̂ = 0.52.

Step 4: Determine the p-value.
Using a one-sample proportion test, we can calculate the probability of observing a proportion of 52% or more when the true proportion is less than or equal to 0.5 (under the null hypothesis). This probability is the p-value.

Step 5: Interpret the results.
If the p-value is less than the predetermined significance level of 0.05, we reject the null hypothesis. If the p-value is greater than or equal to 0.05, we fail to reject the null hypothesis.

b. Based on the result in (a), if the p-value is less than 0.05, we can conclude that the claim implied by the authors is valid. If the p-value is greater than or equal to 0.05, we cannot conclude that more than half of all mobile phone users are now using the mobile Internet.

c. Suppose the survey found that 53% of mobile phone users are now using the mobile Internet. We will repeat the steps from (a) using the new observed proportion of p̂ = 0.53 to test the claim.

Step 1: State the hypotheses.
Null hypothesis (H0): p ≤ 0.5 (less than or equal to half of all mobile phone users are using the mobile Internet)
Alternative hypothesis (Ha): p > 0.5 (more than half of all mobile phone users are using the mobile Internet)

Step 2: Formulate an analysis plan.
We will use a one-sample proportion test to analyze the survey data.

Step 3: Analyze sample data.
Based on the survey, 53% of the 1,132 mobile phone users are using the mobile Internet. This gives us an observed proportion of p̂ = 0.53.

Step 4: Determine the p-value.
Using a one-sample proportion test, we can calculate the probability of observing a proportion of 53% or more when the true proportion is less than or equal to 0.5 (under the null hypothesis). This probability is the p-value.

Step 5: Interpret the results.
If the p-value is less than the predetermined significance level of 0.05, we reject the null hypothesis. If the p-value is greater than or equal to 0.05, we fail to reject the null hypothesis.

d. By comparing the results of (b) and (c), we can determine whether the claim that more than half of all mobile phone users are now using the mobile Internet is valid. If the p-value in (c) is less than 0.05, indicating a significant proportion, then the claim is valid. If the p-value is greater than or equal to 0.05, the claim cannot be supported.

a. To test the hypothesis that more than half of all mobile phone users are now using the mobile internet, we can follow the five-step p-value approach to hypothesis testing:

Step 1: State the null hypothesis (H0) and the alternative hypothesis (Ha).
H0: The proportion of mobile phone users using the mobile internet is 0.5 or less.
Ha: The proportion of mobile phone users using the mobile internet is greater than 0.5.

Step 2: Set the significance level (α), which is given as 0.05 in this case.

Step 3: Collect the sample data and calculate the test statistic.
In this case, the sample proportion is 52% or 0.52. The sample size is 1,132. The test statistic to use is the z-statistic, which follows a standard normal distribution.

The formula for the z-test statistic is:
z = (p - P) / √[(P * (1 - P)) / n]
where p is the sample proportion, P is the hypothesized proportion, and n is the sample size.

Substituting the values:
z = (0.52 - 0.5) / √[(0.5 * (1 - 0.5)) / 1132]
z = 0.02 / √[(0.25) / 1132]
z ≈ 0.02 / 0.01579
z ≈ 1.269

Step 4: Calculate the p-value.
We need to find the p-value corresponding to a z-score of 1.269. Consulting a standard normal distribution table, the p-value is found to be approximately 0.102.

Step 5: Make a decision.
Since the p-value (0.102) is greater than the significance level (0.05), we fail to reject the null hypothesis. We do not have enough evidence to prove that more than half of all mobile phone users are now using the mobile internet.

b. Based on the result in (a), we cannot conclude that the claim implied by the authors is valid. The survey does not provide sufficient evidence to support the claim that more than half of all mobile phone users are now using the mobile internet.

c. Let's repeat the hypothesis test assuming the survey found that 53% of mobile phone users are now using the mobile internet.

H0: The proportion of mobile phone users using the mobile internet is 0.5 or less.
Ha: The proportion of mobile phone users using the mobile internet is greater than 0.5.

Calculating the test statistic:

z = (0.53 - 0.5) / √[(0.5 * (1 - 0.5)) / 1132]
z = 0.03 / √[(0.25) / 1132]
z ≈ 0.03 / 0.01579
z ≈ 1.907

Calculating the p-value corresponding to a z-score of 1.907:

The p-value is approximately 0.028.

Since the p-value (0.028) is less than the significance level (0.05), we reject the null hypothesis. We have enough evidence to support the claim that more than half of all mobile phone users are now using the mobile internet.

d. Comparing the results:

In part (b), when the survey reported 52% of mobile phone users using the mobile internet, we failed to reject the null hypothesis. This means we could not say with confidence that more than half of all mobile phone users are using the mobile internet.

In part (c), when the survey reported 53% of mobile phone users using the mobile internet, we rejected the null hypothesis. This means we have evidence to support the claim that more than half of all mobile phone users are using the mobile internet.

Therefore, the results of parts (b) and (c) differ. The result in part (c) provides stronger evidence to support the claim implied by the authors than the result in part (b).