Mortgage Rates In 2001, the mean contract interest rate
for a conventional 30-year first loan for the purchase of a
single-family home was 6.3 percent, according to the U.S.
Federal Housing Board.A real estate agent believes that
interest rates are lower today and obtains a random sample
of 41 recent 30-year conventional loans. The mean
interest rate was found to be 6.05 percent, with a standard
deviation of 1.75 percent. Is this enough evidence to
conclude that interest rates are lower at the level
of significance?
Take a shot, what do you think?
Hint: no.
To determine whether there is enough evidence to conclude that interest rates are lower at the level of significance, we can perform a hypothesis test.
Step 1: Establish the null and alternative hypothesis.
The null hypothesis (H0) states that there is no significant difference between the mean interest rate in 2001 (μ1) and the mean interest rate today (μ2). In this case, H0: μ1 = μ2.
The alternative hypothesis (Ha) states that there is a significant difference between the mean interest rate in 2001 and the mean interest rate today. In this case, Ha: μ1 > μ2 (since we want to test if interest rates are lower today).
Step 2: Set the level of significance (α).
The level of significance, denoted by α, determines the probability of making a Type I error (rejecting the null hypothesis when it is actually true). Let's assume a significance level of α = 0.05 for this test.
Step 3: Perform the hypothesis test.
We will use a t-test to compare the means of two independent populations. Since the population standard deviation is unknown, we need to use the t-distribution.
The test statistic for this test is given by:
t = (x̄1 - x̄2) / √((s1^2 / n1) + (s2^2 / n2))
where x̄1 and x̄2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
In this case, x̄1 = 6.3, x̄2 = 6.05, s1 = 1.75, s2 = unknown, n1 = unknown, and n2 = 41.
Step 4: Calculate the test statistic and p-value.
Since we don't have the sample standard deviation and sample size from 2001, we cannot calculate the test statistic and p-value in this case.
Step 5: Make a decision.
Since we don't have the test statistic and p-value, we cannot make a decision based on statistical analysis. However, we can make an assumption.
If the mean interest rate of the 41 recent 30-year conventional loans is significantly lower than the mean interest rate in 2001 (6.3%), then we can conclude that there is enough evidence to support the agent's belief that interest rates are lower today.
Please note that without the actual data from 2001, it is difficult to draw a definitive conclusion. It is recommended to collect more data and perform a more rigorous statistical analysis to confirm the findings.