Question

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. Under 25 10 10 11 15 7 11 10 9 Over 60 4 8 7 7 4 5 1 7 4 10 5 1. Find the degrees of freedom for unequal variance test. (Round down answer to nearest whole number.) 2. State the decision rule for 0.01 significance level: H0 μunder ≤ μover; μunder > μover. (Round your answer to 3 decimal places.) 3. Compute the value of the test statistic. (Round your answer to 2 decimal places.) 4. At the .01 significance level, can the bank management conclude that younger customers use the ATMs more?

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u_{Under 25}< u_{Over 60}
Alternative hypothesis: u_{Under 25} > u_{Over 60}

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.01. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

SE = sqrt[(s₁²/n₁) + (s₂²/n₂)]
SE = 1.1174
DF = 18
t_critical = 2.553

Rejection region is t > 2.553
t = [ (x₁ - x₂) - d ] / SE

t = 4.59

where s₁ is the standard deviation of sample 1, s₂ is the standard deviation of sample 2, n₁ is thesize of sample 1, n₂ is the size of sample 2, x₁ is the mean of sample 1, x₂ is the mean of sample 2, d is the hypothesized difference between population means, and SE is the standard error.

The observed difference in sample means produced a t statistic of 4.59.

Therefore, the P-value in this analysis is less than 0.001.

Interpret results. Since the P-value (almost 0) is less than the significance level (0.01), we have to reject the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that younger customers use the ATMs more.