Question

Suppose the average income for a city is estimated to be $45,000, with a standard deviation of $15,000. A recent survey of 50 people in the town had a sample mean income of $42,000. To a confidence level of α = 0.10, test whether the average income has gone down from $45,000.

(a) What is the null hypothesis?

(b) What is the alternative hypothesis?

(c) Which kind of test is this (left-tail, right-tail, or two-tail)?

(d) Find zdata

(e) Find the critical region.

(f) Do we reject the null hypothesis or fail to reject it?

Answer 1

Below are the null and alternative Hypothesis,
a)

Null Hypothesis: μ = 45000

b)

Alternative Hypothesis: μ < 45000

c)

This is left tailed test, for α = 0.1 and df = 49

d)

Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (42000 - 45000)/(15000/sqrt(50))
t = -1.414

e)

Critical value of t is -1.299.
Hence reject H0 if t < -1.299

f)

As the value of test statistic, t is outside critical value range, reject the null hypothesis

Using z

Below are the null and alternative Hypothesis,
a)

Null Hypothesis: μ = 45000

b)

Alternative Hypothesis: μ < 45000

c)

This is left tailed test, for α = 0.1

d)

Test statistic,
z = (xbar - mu)/(sigma/sqrt(n))
z = (42000 - 45000)/(15000/sqrt(50))
z = -1.41

e)
Critical value of z is -1.28.
Hence reject H0 if z < -1.28

f)

As the value of test statistic, z is outside critical value range, reject the null hypothesis