Question

What is the p-value? -- Researcher Jessie is studying how the fear of going to the dentist affects an adult's actual number of visits to the dentist. She asks a random sample of adults whether or not they fear going to the dentist and also how many times they have gone in the past 10 years. She would like to assess if the average number of visits made by adults who fear going to the dentist (Group 1) is higher than the average number of visits for those who don't have that fear (Group 2), that is, test H0: μ1 = μ2 versus Ha: μ1 > μ2, using a 10% significance level. Her random sample of adults resulted in 14 stating they feared going to the dentist and 31 stated they did not fear going to the dentist. The first sample mean was 1.71 pooled standard errors above the second sample mean. Jessie has asked you to provide a complete sketch of the p-value that she can include in her report. You can use the shiny app in R which will give the exact p-value or make a complete sketch by hand and provide the bounds for the p-value using the T table.
2) Jessie also remembers some condition about a normal model required for her two populations of responses. She asks you to check this condition for her. You recall that a QQ plot helps to assess if a population of responses can be considered normally distributed. How many QQ plots do you need to make in this case?

None, since the total sample size of 45 adults is large you can just use the CLT.

One, for the 45 responses (number of visits) reported by the 45 adults that were surveyed.

Two, one for the 14 responses (number of visits) by the adults who fear going to the dentist and one for the 31 responses (number of visits) by the adults who do not fear going to the dentist.

Answer 1

Answer:

Two, one for the 14 responses (number of visits) by the adults who fear going to the dentist and one for the 31 responses (number of visits) by the adults who do not fear going to the dentist.

Step-by-step explanation:

Hello!

1)

You want to test if the average visits to the dentist of people who fear to visit it are greater than the average visits of people that don't fear it.

In this case, the statistic to use is a pooled Student t-test. The reason I've to choose this test is that one of your sample sizes is small (n₁= 14) and the t-test is more accurate for small samples. Even if the second sample is greater than 30, if both variables are normally distributed, the pooled t-test is the one to use.

H₀: μ₁ = μ₂

H₁: μ₁ > μ₂

α: 0.10

t= (X₁[bar]-X₂[bar]) - (μ₁ - μ₂) ~ t$_{n₁+n₂-2}$

        Sₐ√(1/n₁+1/n₂)

Where

X₁[bar] and X₂[bar] are the sample means of both groups

Sₐ is the pooled standard deviation

This is a one-tailed test, you will reject the null hypothesis to big numbers of t. Remember: The p-value is defined as the probability corresponding to the calculated statistic if possible under the null hypothesis (i.e. the probability of obtaining a value as extreme as the value of the statistic under the null hypothesis), and in this case, is also one-tailed.

P(t$_{n₁+n₂-2}$ ≥ t$_{H0}$) = 1 - P(t$_{n₁+n₂-2}$ < t$_{H0}$)

Where t$_{H0}$ is the value of the calculated statistic.

Since you didn't copy the data of both samples, I cannot calculate it.

2)

Well there was one sample taken and separated in two following the criteria "fears the dentist" and "doesn't fear the dentist" making two different samples, so this is a test for two independent samples. To check if both variables are normally distributed you need to make two QQplots.

I hope it helps!