In Exercises 17 and 18, (c) find the standardized test statist...

Question

In Exercises 17 and 18, (c) find the standardized test statistic t, Assume the samples are random and independent, and the populations are normally distributed.

A real estate agent claims that there is no difference between the mean household incomes of two neighborhoods. The mean income of 12 randomly selected households from the first neighborhood is $52,750 with a standard deviation of $2900. In the second neighborhood, 10 randomly selected households have a mean income of $51,200 with a standard deviation of $2225. At α=0.01, can you reject the real estate agent’s claim? Assume the population variances are equal.

Accepted Answer

Hello. In this video, a company wants to compare the average weekly working hours of employees in two departments. In Department X, a sample of 11 employees has a mean of 41.2 hours with a standard deviation of 2.6 hours. In Department Y, a sample of 13 employees has a mean of 39.8 hours with a standard deviation of 3.1 hours. We want to find the standardized test statistic T. And assume equal population variances. So let's go ahead and start this problem by writing down the given information. In department X we have a sample. Of 11 employees. They have a mean working hour of 41.2 hours, and they have a sample standard deviation of 2.6 hours. In the 2nd department, they have a sample size of 13 employees with a mean. Work hours of 39.8 hours. And a sample standard deviation of 3.1 hours. Now in this case here because we are assuming equal population variances and we are testing the claim between two different populations, we are going to use a pooled. Two-tailed Tea test Because we are using a pooled two-tail T test, we can go ahead and review the the definition of the T test statistic for a pooled two-tail T test. So the T test statistic for a pulled two-tail T test is defined. As the mean of the 1st department minus the mean of the 2nd department. Divided by the pooled sample standard deviation multiplied by the square root of 1 divided by the sample size of the first department plus 1 divided by the sample size of the second department. Of course, the T test statistic does require the pool sample vary or sample standard deviation. We have a definition for the square pooled sample standard deviation. That is going to be the first sample size -1. Multiplied by the first sample standard deviation squared. Plus the 2nd sample size minus 1 multiplied by the 2nd sample standard deviation squared and this will all be divided by the degree of freedom. And finally for a pooled two tail t test, the degree of freedom is defined as the sum of the two sample sizes. -2 So in order to find our test statistic, we need to first start by finding the degree of freedom. So, by taking the sum of both of the sample sizes and subtracting two, the degree of freedom will be defined as 11 + 13 minus 2, which is going to give us a degree of freedom of 22. Now that we have the degree of freedom, we can calculate the pooled sta sample standard deviation. Squared So the pooled sample standard deviation squared is going to be defined as 11 minus 1, which is 10, multiplied by the sample standard deviation squared of X, that'll be 2.6 squared, plus the second sample size minus 1, 13 minus 1 is 12, multiplied by the second standard deviation squared, which is 3.1 squared, and this will all be divided by the degree of freedom, 22. Now here, if we were to plug this into a calculator, we will get a squared pole sample standard deviation of approximately 8.3145. But of course, we only need the standard pulled sample standard deviation. So the next thing we're going to do is we're going to take the square root of both sides of this equation. Taking the square root gives us an approximate pool sample standard deviation of 2.8835. Now that we have the polled sample standard deviation, we can calculate our T test statistic. So, the T test statistic is going to be defined. As the first average of department X, which is 41.2 minus the average of department Y, which is 39.8. And this will be divided by the pool sample standard deviation 2.8835 multiplied by the square root of 1 divided by the first sample size, which is 1 divided by 11, plus 1 divided by the second sample size, which is 1 divided by 13. By plugging this into a calculator, this gives us a T test statistic of approximately 1.1851. And if we were to round this to one decimal place, this is going to give us a T test statistic of approximately 1.2. This is going to be the solution to our problem, and the overall solution is going to be option C. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Key Concepts

Two-Sample t-Test for Means

Pooled Variance and Equal Variance Assumption

Significance Level (α) and Hypothesis Testing

Watch next