Question

A sample of 109 mortgages approved during the current year showed that 48 were issued to a single-earner family or individual. The historical percentage is 40 percent.
  
At the .05 level of significance in a right-tailed test, has the percentage of single-earner or individual mortgages risen?
  
(a-1) H₀: ππ ≤ .40 versus H₁: ππ > .40. Choose the right option.
  

1. Reject H₀ if z_calc > 1.645
2. Reject H₀ if z_calc < 1.645

  

• a

• b

(a-2) Calculate the test statistic. (Round intermediate calculations to 2 decimal places. Round your answer to 4 decimal places.)
  
Test statistic            
  
(a-3) The null hypothesis should be rejected.
  

• True

• False

(b) Is this a close decision?
  

• Yes

• No

  
(c) State any assumptions that are required.
  

1. n ππ < 10 and n(1 − ππ ) > 10
2. n ππ < 10 and n(1 − ππ ) < 10
3. n ππ > 10 and n(1 − ππ ) > 10
4. n ππ > 10 and n(1 − ππ ) < 10

  

• a

• b

• c

• d

Answer 1

Given that, n = 109 and x = 48

=> sample proportion = 48/109 = 0.4404

a-1) The null and alternative hypotheses are,

H0 : Π ≤ 0.40 versus H1 : π > 0.40

This test is right-tailed test.

critical value at 0.05 level of significance is, z^* = 1.645

Decision Rule : Reject H0 if Zcalc > 1.645

a-2) Test statistic is,

=> Test statistic = z = 0.8610

a-3) Since, test statistic = 0.8610 < 1.645, we fail to reject the null hypothesis.

Answer : False

b) No, this is not close decision.

c) Required assumptions are,

nπ > 10 and n(1 - π) > 10