The Commercial Bank and Trust Company is studying the use of...

The Commercial Bank and Trust Company is studying the use of its automatic teller machines (ATMs). Of particular interest is whether young adults (under 25 years) use the machines more than senior citizens. To investigate further, samples of customers under 25 years of age and customers over 60 years of age were selected. The number of ATM transactions last month was determined for each selected individual, and the results are shown below. At the .01 significance level, can bank management conclude that younger customers use the ATMs more?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. In this context, the procedure involves setting up a null hypothesis (that there is no difference or that senior citizens and younger adults have equal ATM usage) against an alternative hypothesis (that younger customers use the ATMs more frequently).

Significance Level

The significance level, often represented as alpha (?), is the threshold for rejecting the null hypothesis. At a 0.01 significance level, there is a 1% risk of rejecting the null hypothesis when it is in fact true (Type I error), ensuring a highly stringent test for evidence against the null hypothesis.

One-Tailed Test

A one-tailed test is used when the research hypothesis predicts a specific direction of difference. In this case, the hypothesis that younger adults use ATMs more than senior citizens requires a one-sided approach, testing for an increase in usage specifically in one direction rather than any difference.

Two-Sample Comparison

A two-sample comparison involves analyzing data collected from two independent groups to determine if there is a statistically significant difference between their means. This approach is relevant when comparing the ATM usage of younger adults to that of senior citizens.

T-Test for Independent Samples

The t-test for independent samples is a statistical test used to compare the means of two independent groups, particularly when population variances are unknown and sample sizes are relatively small. This test helps determine whether any observed difference in ATM usage between the two age groups is statistically significant or merely a result of random variation.

Transcript

00:01 So at a 1 % significance level, we want to see if the mean number of transactions made at the atm is equal for younger people versus older.

00:12 And younger is under 25 and older is over 60.

00:16 And alternately, that the younger group will use the atms more.

00:21 And so we have a sample size for the younger group of 8.

00:25 8.

00:27 For our older group, we have a sample size of 11.

00:32 And the mean of our first group, the younger group, is 10 .375 transactions with a sample standard deviation of 2 .2638.

00:49 And our older group had a mean number of uses of 5 .636 those are repeating and the standard deviation for that group is 2 .4606 and for our test we will be doing a two sample t test because our sample sizes are so small and we will be assuming that these two standard deviations are equal.

01:23 So that causes us to have to find our pooled p, or our, excuse me, our pooled variance.

01:29 And that pooled variance will end up having a size of seven times, one less than the first sample size times the variance of the first group.

01:39 And then one less than this sample size times the variance of this group.

01:46 And then we'll take eight plus 11 less two.

01:51 So that will give us our degrees of freedom.

01:53 And we can see that that's going to be 17.

01:55 So our degrees of freedom is 17.

01:59 And let's get this calculation.

02:00 And then let's find our test statistic and then draw our little table and find out whether we're going to be having evidence against the null or not...

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