Question

A researcher is interested in seeing if the average income of rural families is different than that of urban families. To see if his claim is correct he randomly selects 60 families from a rural area and finds that they have an average income of $68428 with a population standard deviation of $775. He then selects 59 families from a urban area and finds that they have an average income of $69066 with a population standard deviation of $924. Perform a hypothesis test using a significance level of 0.01 to test his claim. Let rural families be sample 1 and urban familis be sample 2.

The correct hypotheses are:

• H0:μ1≤μ2H0:μ1≤μ2
  HA:μ1>μ2HA:μ1>μ2(claim)
• H0:μ1≥μ2H0:μ1≥μ2
  HA:μ1<μ2HA:μ1<μ2(claim)
• H0:μ1=μ2H0:μ1=μ2
  HA:μ1≠μ2HA:μ1≠μ2(claim)

Since the level of significance is 0.01 the critical value is 2.576 and -2.576

The test statistic is: (round to 3 places)

The p-value is: (round to 3 places)

The decision can be made to:

• reject H0H0
• do not reject H0H0

The final conclusion is that:

• There is enough evidence to reject the claim that the average income of rural families is different than that of urban families.
• There is not enough evidence to reject the claim that the average income of rural families is different than that of urban families.
• There is enough evidence to support the claim that the average income of rural families is different than that of urban families.
• There is not enough evidence to support the claim that the average income of rural families is different than that of urban families.

Answer 1

Solution:

Given:

Claim: the average income of rural families is different than that of urban families.

Sample 1 rural area:

n1 = 60

Sample 2 urban area:

n2 = 59

significance level = 0.01

The correct hypotheses are:

H0:μ1=μ2 Vs HA:μ1≠μ2 (claim)

Since the level of significance is 0.01 the critical value is 2.576 and -2.576

for two tailed test, left tail area = 0.01 / 2 = 0.005

Use excel command:

=NORM.S.INV(0.005)

=-2.576

Thus the critical value is 2.576 and -2.576.

The test statistic is:

The p-value is:

Use following Excel command:

=2*NORM.S.DIST(z,cumulative)

=2*NORM.S.DIST(-4.078,TRUE)

= 0.0000454

= 0.000

Thus p-value is 0.000

The decision can be made to:

Decision Rule:
Reject null hypothesis H0, if P-value < 0.01 level of significance, otherwise we fail to reject H0
Since p-value is 0.000 < 0.01 level of significance, we reject null hypothesis H0.

Thus

The decision can be made to: reject H0

The final conclusion is that:

There is enough evidence to support the claim that the average income of rural families is different than that of urban families.