Question

Asked Apr 4, 2019

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Lydia Hayward is an external research consultant hired by FirstCard, a major credit card company, to find evidence that the population mean total credit card balance for adults ages 25 to 34 with a college degree in the region is greater than for adults ages 25 to 34 without a college degree. Due to research already conducted by FirstCard, Lydia assumes that the population standard deviation is $2,047.24 for adults ages 25 to 34 with a college degree and $1,749.65 for those without a college degree. She conducts a survey at a local shopping center and selects every tenth adult ages 25 to 34 with a college degree and every tenth adult ages 25 to 34 without a college degree to ask for the total credit card balance. The results of the samples are shown in the table below. Explain whether a hypothesis test for the difference between two means of independent samples is appropriate, and if so, determine the null and alternative hypotheses for this hypothesis test. Let μ1be the population mean total credit card balance for adults ages 25 to 34 with a college degree and μ2 be the population mean total credit card balance for adults ages 25 to 34 without a college degree.

**With College Degree****Without College Degree**

x⎯⎯⎯1=$6,312.91

x⎯⎯⎯2=$5,483.15

n1=271

n2=315

Step 1

**Given information:**

Denote the population mean total credit card balance for adults 25 to 34 with a college degree as μ_{1} and the population mean total credit card balance for adults 25 to 34 without a college degree as μ_{2.}

The population standard deviation in total credit card balance for adults 25 to 34 with a college degree is σ_{1} = $2,047.24 and the population standard deviation in total credit card balance for adults 25 to 34 without a college degree is σ_{2} = $1,749.65.

*Independent samples:*

Any two randomly selected samples are said to be independent, if the observations in one sample do not depend on the observations in the other sample. In other words it can be said that the observations in one sample should not influence the observations in the other sample.

Here, Lydia selects every tenth person with a college degree and every tenth person without a college degree at a shopping mall and asks them their respective total credit card balance. That is, the two samples are selected randomly without any bias.

In general it is known that the adults with a college degree and the adults without a college degree are independent of each other.

That is, the credit card total balance of adults with a college degree does not depend on the credit card total balance of adults without a college degree. That is, observations in the sample adults with a college degree do not influence the observation in the sample adults without a college degree.

Therefore, the two samples are independent samples.

The sample size of adults 25 to 34 with a college degree is *n*_{1} = 271.

The sample size of adults 25 to 34 without a college degree is *n*_{2} = 315.

*Assumptions for two independent samples z test:*

The required conditions for using two independent samples *z *test are given below:

- The two samples must be randomly selected from the two populations.
- The two samples must be independent of one another.
- Population standard deviations must be known.
- Either each sample sizes (
*n*) must be greater than 30 or the populations must be normally distributed.

Here, the samples are randomly selected, the samples are independent of each other. The population standard deviations are known and the samples are sufficiently large.

Hence, all the assumptions are satisfied.

The test for the difference between two means of independent samples is valid.

Step 2

**Test hypothesis:**

The objective is to test, whether or not the population mean total credit card balance for adults 25 to 34 with a college degree is greater than the population mean total credit card balance for adults 25 to 34 without a college degree.

*The hypotheses are given below:*

*Null hypothesis:*

*H*0 : μ1 = μ2

That is, the population mean total credit card balance for adults 25 to 34 with a college degree is equal to the population mean total credit card ...

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