Reformulate your hypothesis test from your week 5 discussion to incorporate a 2-

Question

Reformulate your hypothesis test from your week 5 discussion to incorporate a 2-sample hypothesis test, as specified in Chapter 10. What would be your data? What is your null hypothesis? What is your alternate hypothesis? What would be your Type 1 and Type 2 errors relative to your decision? Suppose you have a p-value of 0.01, what does this mean relative to your problem and decision? Suppose the p-value is 0.20, what does this mean relative to your problem and decision? If you reformulated your design for 3 or more samples, what would be the implications of interaction? When would you use Tukey-Kramer test?

This is week 5 answer to help with the above question:

Hypothesis Test

Suppose we are interesting in checking the claim whether the younger customers use the ATMs more than the older customers and for this purpose we need to collect some data for checking our claim by using the hypothesis test. So, we collect the data for the use of ATMs by the younger and older customers. Suppose, we collected the data for the 8 younger customers and 11 older customers who use the ATMs for transactions and after collecting this data we have to analyse this data for checking the claim we interested in. We defined the null and alternative hypothesis for this test as below:

Null hypothesis: H0: There is no any significant difference in the use of ATMs by younger and older customers.

Alternative hypothesis: Ha: The younger customers use the ATMs more than the older customers.

We assume the level of significance or the value for alpha as 0.01 to 1%.

For the sample of younger customers, we get the mean = 10.375 and SD = 2.2638 while for the sample of older customers, we get the mean = 5.6363 and SD = 2.4606

The test statistic formula is given as below:

Test statistic = t = ( sample 1 mean - sample 2 mean) / sqrt [(var1/n1)+(var2/n2) ]   

where n1 and n2 are the sample sizes and var1 is the sample variance for first sample and var2 is the variance for second sample. we are given n1 = 8 and n2 =11.

now, plug all the values in the above formula and calculate the values for test statistic.

The test is given as below:

Separate-Variances t Test for the Difference Between Two Means

(assumes unequal population variances)

Data

Hypothesized Difference

0

Level of Significance

0.01

Population 1 Sample

Sample Size

8

Sample Mean

10.375

Sample Standard Deviation

2.2638

Population 2 Sample

Sample Size

11

Sample Mean

5.636363636

Sample Standard Deviation

2.4606

Intermediate Calculations

Numerator of Degrees of Freedom

1.4186

Denominator of Degrees of Freedom

0.0889

Total Degrees of Freedom

15.9526

Degrees of Freedom

15

Standard Error

1.0913

Difference in Sample Means

4.7386

Separate-Variance t Test Statistic

4.3420

Upper-Tail Test

Upper Critical Value

2.6025

p-Value

0.0003

Reject the null hypothesis

Here, we get the p-value as 0.0003 which is less than the given level of significance or alpha value 0.01, so we reject the null hypothesis that there is no any significant difference in the use of ATMs by younger and older customers. This means, we conclude that the younger customers use the ATMs more than the older customers.

Here, type I and type II error are given as below:

Type I error is the probability of rejecting the null hypothesis that there is no significant difference in the use of ATMs by younger customers and older customers while actually there is no significant difference.

Type II error is the probability of do not rejecting the null hypothesis when actually there is a significant difference in the use of ATMs by the younger and older customers.

Separate-Variances t Test for the Difference Between Two Means

(assumes unequal population variances)

Data

Hypothesized Difference

0

Level of Significance

0.01

Population 1 Sample

Sample Size

8

Sample Mean

10.375

Sample Standard Deviation

2.2638

Population 2 Sample

Sample Size

11

Sample Mean

5.636363636

Sample Standard Deviation

2.4606

Intermediate Calculations

Numerator of Degrees of Freedom

1.4186

Denominator of Degrees of Freedom

0.0889

Total Degrees of Freedom

15.9526

Degrees of Freedom

15

Standard Error

1.0913

Difference in Sample Means

4.7386

Separate-Variance t Test Statistic

4.3420

Upper-Tail Test

Upper Critical Value

2.6025

p-Value

0.0003

Reject the null hypothesis

Explanation / Answer

Null hypothesis: H0: There is no any significant difference in the use of ATMs by younger and older customers.

Alternative hypothesis: Ha: The younger customers use the ATMs more than the older customers.

alpa:1%

Test statistic:

t=( sample 1 mean - sample 2 mean) / sqrt [(var1/n1)+(var2/n2) ]

t=4.34

conclusion:

P=0.0003 <0.01 we reject the H0. Hence the claim is significant

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