Question

A random sample of 100 observations from a quantitative population produced a sample mean of 29.8 and a sample standard deviation of 7.2. Use the p-value approach to determine whether the population mean is different from 31. Explain your conclusions. (Use α = 0.05.)

State the null and alternative hypotheses. (Choose Correct Letter)

(a) H₀: μ = 31 versus H_a: μ < 31

(b) H₀: μ ≠ 31 versus H_a: μ = 31    

(c) H₀: μ < 31 versus H_a: μ > 31

(d) H₀: μ = 31 versus H_a: μ > 31

(e) H₀: μ = 31 versus H_a: μ ≠ 31

Find the test statistic and the p-value. (Round your test statistic to two decimal places and your p-value to four decimal places.)

z	=
p-value	=

State your conclusion. (Choose Correct Letter)

(a) The p-value is greater than alpha, so H₀ is not rejected. There is sufficient evidence to indicate that the mean is different from 31.

(b) The p-value is less than alpha, so H₀ is rejected. There is insufficient evidence to indicate that the mean is different from 31.    

(c) The p-value is greater than alpha, so H₀ is not rejected. There is insufficient evidence to indicate that the mean is different from 31.

(d) The p-value is less than alpha, so H₀ is rejected. There is sufficient evidence to indicate that the mean is different from 31.

Answer 1

Given :-

Sample Size n = 100

= 29.8 Sample mean

S = 7.2 sample standard deviation

= 31 population mean

null and alternative hypotheses

Null hyothesis H₀ =

Alternative hypothesis H₁ =

Part a) Correct option is (e).

Part b) Find the test statistic

Since we have sample standard deviation we need to use t test, if we would have population standard deviation () we would have use Z test.

Test Statistic :-

t =

t = (29.8 - 31 ) / ( 7.2 / )

t = -1.67

Test Criteria :-

Reject Null hypothesis if | t | > t_/2,n-1

t_/2,n-1 = t_0.025,99 = 1.984

| t | > t_/2,n-1 = | -1.67 | > 1.984 = 1.67 < 1.984, hence we fail to reject null hypothesis

Decision based on P value

P value = 0.0987

Test Criteria :- Reject null hypothesis if P value < = 0.05 (Level of Significance)

0.0987 > 0.05, which does not satisfy the condition of P value < = 0.05 , hence we fail to reject null hypothesis.

Comclusion :- Accpet Null hypothesis i.e

For conclusion, correct option is (a).