Question

The belief is that the mean number of hours per week of part-time work of high school seniors in a city is 10.5 hours. Data from a simple random sample of 23 high school seniors indicated that their mean number of part-time work was 11.5 with a standard deviation of 1.4. Test whether these data cast doubt on the current belief. (use α = 0.05)

Required:
a. State your null and alternative hypotheses.
b. Sketch the rejection region.
c. Calculate the test statistic. Plot this value in your sketch in part b.
d. Determine the P-value for your test.
e. State your conclusions clearly in complete sentences.

Answer 1

a) See step by step explanation

b) See annex

c) t(s) = 3,4255

d) p- value = 0,00146 or 0,15 %

e) See step by step explanation

Explanation:

As n < 30 we use a t-student distribution

Population mean μ₀ = 10,5

Sample size n = 23

Degree of freedom n - 1 = 22

Sample mean μ = 11,5

Sample standard deviation s = 1,4

Confidence Interval 95 %

a) Null Hypothesis H₀ μ = μ₀

Alternative Hypothesis Hₐ μ > μ₀

As CI = 95 % α = 5% or α = 0,05

We are solving a one tail-test

With df = 22 and α = 0,05 in t-table we find t(c) = 1,7171

c) t(s) = ( μ - μ₀ ) / s / √n

t(s) = ( 11,5 - 10,5 ) / 1,4 / √23

t(s) = 1 * 4,7958 / 1,4

t(s) = 3,4255

We compare t(s) and t(c)

t(s) > t(c) and t(s) is in the rejection region

Then we reject the null hypothesis.

d) P-value for t(s) = 3,4255 is from t-table equal to:

We find for df = 22 α = 0,001 and α = 0,005

values of t

t 3,505 2, 819 Δt = 0,686

α 0,001 0,005 Δα = 0,004

with these values we interpolate by rule of three

0,686 ⇒ 0,004

(3,505 - 3,4255) ⇒ x

x = 0,000463

and P-value = 0,00146 or 0,15 %

e) The p-value indicates we are far away to consider the accptance of H₀