Question

Simple random samples of high-interest mortgages and low-interest mortgages were obtained. For the 15 high-interest mortgages, the borrowers had a mean credit score of 569 and a standard deviation of 40. For the 40 low-interest mortgages, he borrowers had a mean credit score of 585 and a standard deviation of 24. Test the claim that the mean credit score of borrowers with high-interest mortgages is lower than the mean credit score of borrowers with low-interest mortgages at the 0.10 significance level.

Claim:

Opposite:

The test is:

The test statistic is: t=

The critical Value is:

Based on this we?

Conclusion?

Answer 1

X = 569 X2 = 585 Also, the provided sample standard deviations are: Si = 40 S2 = 24 and the sample sizes are ni 15 and n2 - 40. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: Mi = M2 Ha: M1 < u2 This corresponds to a left-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used. (2) Rejection Region Based on the information provided, the significance level is a = - 0.10, and the degrees of freedom are df = 53. In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal: – 1.298, for Hence, it is found that the critical value for this left-tailed test is te a = 0.10 and df 53. = The rejection region for this left-tailed test is R= = {t:t< -1.298}. (3) Test Statistics Since it is assumed that the population variances are equal, the t-statistic is computed as follows: t= X1 - X2 (ni-1)s? +(n2-1)s ni+n2-2 + ni n2 569 – 585 1.816 (15–1)402 +(40-1)242 ( its + ho 15+40-2 (4) Decision about the null hypothesis Since it is observed that t = -1.816 <te = - -1.298, it is then concluded that the null hypothesis is rejected. = 0.0375 < Using the P-value approach: The p-value is p = 0.0375, and since p 0.10, it is concluded that the null hypothesis is rejected. (5) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean ui is less than pa, at the 0.10 significance level.

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