Question

Smartphone at home ~ Interested in the proportion of people who use smartphones as their primary means of online access at home, Pew Research used representative surveys to determine that 20% of U.S. adults did not use broadband internet at home but did own smartphones in 2018.

Carlos works for a communications company that provides broadband internet to rural areas in the Midwest. After reading the report from Pew Research, Carlos wants to determine if residents in the area his company serves use only their smartphones to access the internet at home at a higher rate than the percentage reported by Pew Research in 2018. He collects a random sample of 455 residents and asks them the following question: "Do you not have broadband internet and use only your smartphones to access the internet at home?"
To correctly describe how the test could be conducted, Carlos can set up the null hypothesis model for this situation by taking 100 cards. He can use green cards to represent the proportion of residents who say that they do not have broadband internet and use only their smartphones to access the internet at home, and white cards to represent the proportion who do not fall in this category. Carlos should shuffle the cards and draw a sample of 455 cards, recording the number of green cards (p_stm). He should repeat this process many, many times and plot the resulting sample proportions to construct the null distribution.
If Carlos finds that 12 of the 3000 simulations fell more than 4.84% below the center of the null distribution, and 25 of the 3000 simulations fell more than 4.84% above the center of the null distribution, he can use these results of the simulation to calculate the p-value for his test.
p-value =

Chip works for a similar communications company that services rural areas in the South. Using the same hypothesis test, Chip uses survey results from his area of the country and calculates a p-value of 0.1002. Based on his p-value of 0.1002, Chip has weak evidence that the null model is a good fit for his observed results.

Answer 1

Carlos and Chip are using hypothesis testing to determine if the proportion of residents using smartphones as the primary online access method at home differs from national data. A p-value of 0.1002 suggests weak evidence against the null hypothesis.

Step-by-step explanation:

To investigate the proportion of individuals using smartphones as their primary means of online access at home, hypothesis testing can be utilized. When Carlos collects data from a sample of 455 residents, he is looking to see if this proportion is higher in his rural Midwestern area than the national average found by Pew Research. Under the null hypothesis, it is assumed that there is no difference between the local and national proportions.

To conduct the test, Carlos can perform simulations using the national proportion as the expected proportion and then use the results of the simulation to calculate the p-value. The p-value helps determine if the observed sample proportion is statistically significantly different from the national proportion. In a similar study, when Chip finds a p-value of 0.1002, it suggests that there is weak evidence against the null hypothesis and that the null model fits his data reasonably well, meaning there is not a significant difference in the usage patterns in his southern rural area.

When conducting such tests, it is also crucial to consider the level of significance, usually set at 5 percent. Rejecting the null hypothesis implies that there is sufficient evidence to support the claim that there is a significant difference between the proportions being compared. If the null is not rejected, it is determined that there is not enough evidence to support that claim.