Question

A researcher 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.

She has asked you to provide a complete sketch of the p-value that she can include in her report. Calculate the degrees of freedom and test-statistic value.

Answer 1

Solution

Let

X = number of visits made by adults of Group 1 [‘fear’ group]

Y = number of visits made by adults of Group 2 [‘non-fear’ group]

Let (µ₁, σ₁) and (µ₂, σ₂) be the mean and SD of X and Y respectively.

Claim:

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)

Hypotheses:

Null: H₀: µ₁ = µ₂ Vs Alternative: H_A: µ₁ > µ₂ [claim]

Test Statistic:

t = (Xbar - Ybar)/SE

where

SE = pooled standard error = [s√{(1/n₁) + (1/n₂)}] and

s² = {(n₁ – 1)s₁² + (n₂ – 1)s₂²}/(n₁ + n₂ – 2);

Xbar and Ybar are sample averages and s₁,s₂ are sample standard deviations based on n₁ observations on X and n₂ observations on Y respectively.

Calculations

Summary of Excel calculations is given below:

Given

‘The first sample mean was 1.71 pooled standard errors above the second sample mean.’

=> Xbar = Ybar + 1.71SE

Or, (Xbar - Ybar) = 1.71SE

And hence,

Test statistic, t = 1.71. Answer 1

Distribution, Significance Level, α, and p-value:

Under H₀, t ~ t_{n1 + n2 - 2}. So, degrees of freedom = (n₁ + n₂ – 2) = 14 + 31 – 2 = 43 Answer 2

Hence,

p-value = P(t_{n1 + n2 - 2} > tcal)

= P(t₄₃ > 1.71)

= 0.0472[Using Excel Function: Statistical TDIST] Answer 3

Decision:

Since p-value < α[given to be 0.1 i.e., 10%], H₀ is rejected.

Conclusion:

There is sufficient evidence to suggest that the claim is valid.

Hence we conclude that 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). Answer 4

DONE