# Math 533

**Topics:**Statistical hypothesis testing, Statistics, Statistical inference

**Pages:**10 (2101 words)

**Published:**May 1, 2013

AJ Davis is a department store chain, which has many credit customers and want to find out more information about these customers. AJ Davis has complied a sample of 50 credit customers with data selected in the following variables: Location, Income (in $1,000’s), Size (Number of people living in the household), Years (number of years the customer has lived in the current location), and Credit Balance (customers current credit card balance on the store’s credit car, in $).

The manager at AJ Davis has speculated the following:

a. The average (mean) annual income was less than $50,000. b. The true population proportion of customers who live in an urban area exceeds 40% c. The average (mean) number of years lived in the current home is less than 13 years d. The average (mean) credit balance for suburban customers is more than $4300

I will analyze the speculated data listed above by performing hypothesis test for each of the above situations (using the Seven elements of a Test Hypothesis with a=.05) in order to see if there is evidence to support my manager’s beliefs in each case (a-d), explain my conclusion in simple terms, compute the p-value with the interpretation, follow up with computing 95% confidence intervals for each of the variables described in a. to d. along with interpreting these intervals. This paper will also include an Appendix with all the steps in hypothesis testing, as well as the confidence intervals and Minitab output

In order to understand how hypothesis testing is done it is important that you know the elements of the Test of Hypothesis, and what each step means.

The Seven elements of a Test of Hypothesis are:

1. Null Hypothesis - A theory about the specific values of one or more population parameters. The theory generally represents the status quo, and we accept it until proven false.

2. Alternative (research) hypothesis (Ha)- A theory about the specific values of one or more population parameters. The theory generally represents the status quo, and we accept it until proven false

3. Test statistic - A sample statistic used to decide whether to reject the null hypothesis.

4. Rejection Region - The numerical values of the test statistic for which the null hypothesis will be rejected.

5. Assumptions- Clear statements of any assumptions made about the populations being sampled.

6. Experiment and calculation of test statistics- Performance of the sampling experiment and determination of the numerical value of the test statistic.

7. Conclusion-

a. If the numerical value of the test statistic falls in the rejection region then we reject the null hypothesis and conclude that the alternative is true. b. If the test statistic does not fall in the rejection region, then we do not reject H0 as we have insufficient data to do so.

a. The average (mean) annual income was less than $50,000

* I found that the average annual incomes are 43.74 or $46,060, and the standard deviation to be 14.64 or $14.064.

* Set up Hypothesis Test

* Ho: µ =50

* H1: µ <50

* For a= 0.5 and “<” in the Ha, I found that z= -1.645, so the “Rejection Region” would be z<-1.645

* Next I calculated the test statistic, using the formula below to calculate the test statistic z.

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where is the mean in the null hypothesis and = s/

* Z= (43.74-50)/7.0711= 2.08, because =-2.07,because =

14.64/ = 7.07107

* The p-value= 0.001. The p-value is another complementary and equally valid way we can evaluate the null and alternative hypotheses is by looking at the p-value and compare the p-value to alpha. If the p-value is less than alpha, reject the null hypothesis and accept the alternative hypothesis, at the given alpha. When you look at the calculated test statistics results you can see that both the test statistic and the...

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