Question

A bank loan officer claims that the mean monthly mortgage payment made by all home owners in a certain

city is $1365. A housing magazine wanted to test this claim. A random sample of 40 homeowners taken by this magazine produced the mean monthly mortgage of $1489 with a standard deviation of $278. Testing at the 2% significance level, would you conclude that the mean monthly mortgage payment made by all homeowners in this city is different from $1365?

      a. State H₀ and H_a symbolically.

      b. Compute the test statistic.

    

     c. Find the p-value.

     d.   State the initial conclusion.

     e. State the conclusion in terms of the original claim.

    

Answer 1

Solution:

a) The null and alternative hypotheses are as follows:

H_{​​​​​​0} : μ = $1365

H_{​​​​​1} : μ ≠ $1365

b) To test hypothesis we shall use one sample t-test for mean. The test statistic is given as follows:

Where, x̄ is sample mean, μ is hypothesized value of population mean, s is sample standard deviation and n is sample size.

We have, x̄ = $1489, μ = $1365, s = $278 and n = 40

The value of the test statistic is 2.82102.

c) Since, our test is two-tailed test, therefore we shall obtain two-tailed p-value for the test statistic. The two-tailed p-value is given as follows:

p-value = 2.P(t > value of the test statistic)

p-value = 2.P(t > 2.82102)

Using R software we get,

p-value = 0.0075

The p-value is 0.0075.

d) Initial conclusion:

We make decision rule as follows:

If p-value is greater than the significance level, then we fail to reject the null hypothesis (H_{​​​​​​0}) at given significance level.

If p-value is less than the significance level, then we reject the null hypothesis (H_{​​​​​​0}) at given significance level.

We have, p-value = 0.0075 and significance level = 2% = 0.02

(0.0075 < 0.02)

Since, p-value is less than the significance level of 0.02, therefore we shall reject the null hypothesis (H_{​​​​​​0}) at 0.02 significance level.

e) Conclusion : At 2% significance level, there is sufficient evidence to conclude that the mean monthly mortgage payment made by all homeowners in this city is different from $1365.

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