Question

A store owner claims the average age of her customers is 32 years. She took a survey of 34 randomly selected customers and found the average age to be 35.7 years with a standard error of 1.631. Carry out a hypothesis test to determine if her claim is valid.

(a) Which hypotheses should be tested?

H₀: μ = 35.7 vs. H_a: μ ≠ 35.7

H₀: μ = 32 vs. H_a: μ ≠ 32    

H₀: μ = 32 vs. H_a: μ > 32

H₀: p = 32 vs. H_a: p ≠ 32

(b) Find the test statistic:  (Use 4 decimals.)

(c) What is the P-value?  (Use 4 decimals.)

(d) What should the store owner conclude, for α = 0.05?

Reject the initial claim of 32 years. There is insufficient evidence the mean customer age is different than 32.

Do not reject the initial claim of 32 years. There is sufficient evidence the mean customer age is different than 32.    

Do not reject the initial claim of 32 years. There is insufficient evidence the mean customer age is different than 32.

Reject the initial claim of 32 years. There is sufficient evidence the mean customer age is different than 32.

(e) If mean customer age really is equal to 32 years, but you conclude it is different than 32 years, which type of error did you make, if any?

Type I error

The p-value is correct; therefore no error was made    

Type II error

Both Type I and Type II error

Answer 1

(a) H₀: μ = 32 vs. H_a: μ ≠ 32  

Null hypothesis states that the average age of the customers is 32 years.

(b) n= 34

sample mean = 35.7

standard error = 1.631

s= 9.5103

Assuming that the data is normally distributed. Also since the Sample size is small and population deviation is not given, we calculate t statistics.  

The t-critical values for a two-tailed test, for a significance level of α=0.05

tc​=−2.035 and tc​=2.035

Since the T statistics falls in the rejection area, we reject the Null hypothesis.

(c) P Value

P value = TDIST (t statistics, df, 2) = TDIST(2.2685, 33, 2)= 0.02997

level of significance = 0.05

Since the P value is less than level of significance we will reject the Null hypothesis

(d) Reject the initial claim of 32 years. There is sufficient evidence the mean customer age is different than 32.

(e) Type I error (rejecting the Null hypothesis when it is true)