statistical math question?

Question

A recent article in the Wall Street Journal reported that the 30-year mortgage rate is now less than 6 percent. A sample of eight small banks in the Midwest revealed the following 30- year rates (in percent):

4.8 5.3 6.5 4.8 6.1 5.8 6.2 5.6
At the .01 significance level, can we conclude that the 30-year mortgage rate for small banks is less than 6 percent? Estimate the p-value.

Answer 1

I have the notes below on how to do this by hand. the quick solution is to use software like R.

> data <- c(4.8, 5.3, 6.5, 4.8, 6.1, 5.8, 6.2, 5.6)
> t.test(data, NULL, "less", 6)

One Sample t-test

data: data
t = -1.6157, df = 7, p-value = 0.07509
alternative hypothesis: true mean is less than 6
95 percent confidence interval:
-Inf 6.062557
sample estimates:
mean of x
5.6375

the p-value of 0.07509 is larger than the significance level and therefore we conclude the null hypothesis is plausible. It is plausible the mean mortgage rate is greater than or equal to 6%.


/// ==== notes === \\\

Hypothesis Test for mean:

assuming you have a large enough sample such that the central limit theorem holds and the mean is normally distributed then to test the null hypothesis H0: μ = Δ

find the test statistic z = (xBar - Δ) / (sx / sqrt(n))

where xbar is the sample average
sx is the sample standard deviation
n is the sample size

The p-value of the test is the area under the normal curve that is in agreement with the alternate hypothesis.

H1: μ > Δ; p-value is the area to the right of z
H1: μ < Δ; p-value is the area to the left of z
H1: μ ≠ Δ; p-value is the area in the tails greater than |z|

for a small sample test for the mean every thing is the same save the test statistic is a t statistic with n - 1 degrees of freedom. However, in this case the underlying distribution must be normal for this test to be valid.

Answer 2

No.

xbar = (4.8+5.3+6.5+4.8+6.1+5.8+6.2+5.6)/8 = 5.6375

sd = 0.6345696

t = (xbar-mu)/(sd/sqrt(n)) = -1.6157

Compare to a t-distribution with 7 degrees of freedom:

p = 0.07509 (one-sided)

p>0.01 so cannot reject the null hypothesis that mortgage rate is >= 6 percent.

Answer 3

all you half to do is add them up and divide by the total number.