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Topic 6 Homework (Nonadaptive) Question 8 of 11 (1 point)|Question Attempt: 1 of Unlimited

Dennis

1 2 3 4 5 6 7 8 9 10 11 Question 8 The university data center has two main servers. The manager of the center wants to examine whether server 1 is receiving tasks which require different processing times from those of server 2. He selects a random sample of 12 tasks from server 1 and a random sample of 13 tasks from server 2. The two samples were chosen separately, so that the selection of the tasks from server 1 did not affect the selection of the tasks from server 2. The following table gives the data, sample means, and sample standard deviations. Processing times in seconds Sample mean Sample standard deviation Server 1 45.60, 45.79, 34.61, 56.14, 37.56, 41.94, 38.80, 30.87, 42.01, 31.26, 43.50, 36.78 40.405 7.071 Server 2 37.24, 33.68, 38.91, 40.55, 42.97, 38.38, 32.16, 38.16, 32.80, 36.24, 58.57, 31.12, 30.06 37.757 7.351

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Send data to Excel (a)Are the two samples of tasks independent or paired (dependent) samples? Choose the best answer. independent samples paired (dependent) samples

(b)The manager of the center wants to use the data to determine whether the mean processing times of server 1 and server 2 differ. Assume that both populations are approximately normally distributed and that the population variances are equal. Choose the appropriate test statistic for the hypothesis test that would be performed. Then calculate that statistic. Carry your intermediate computations to three or more decimal places. Round your answer to three decimal places. (If necessary, consult a list of formulas.)

=z

=t

It is unclear which test statistic to use.

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Solution

Let's break down your problem and approach each part carefully.

(a) Are the two samples independent or paired (dependent) samples?

  • Independent samples: The tasks selected from server 1 do not affect the tasks selected from server 2. Since the two samples were chosen separately, these are independent samples.

Answer: Independent samples.

(b) Hypothesis Test for Comparing Means

The manager wants to test if the mean processing times of the two servers differ. Given the assumptions that:

  1. The populations are approximately normally distributed.
  2. The population variances are equal.

We can use a two-sample t-test for independent samples since we have two independent groups with equal variances.

Steps to Solve:

  1. Hypotheses:

    • Null hypothesis H0H_0: μ1=μ2\mu_1 = \mu_2 (The mean processing times are the same for both servers).
    • Alternative hypothesis HaH_a: μ1μ2\mu_1 \neq \mu_2 (The mean processing times are different for the two servers).
  2. Test Statistic: For equal variances, the test statistic is given by the formula for the two-sample t-test: t=Xˉ1Xˉ2sp1n1+1n2t = \frac{\bar{X}_1 - \bar{X}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} Where:

    • Xˉ1\bar{X}_1 and Xˉ2\bar{X}_2 are the sample means for server 1 and server 2.
    • n1=12n_1 = 12, n2=13n_2 = 13 are the sample sizes.
    • sps_p is the pooled standard deviation given by: sp=(n11)s12+(n21)s22n1+n22s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}} where s1=7.071s_1 = 7.071 and s2=7.351s_2 = 7.351 are the sample standard deviations.

Step-by-Step Calculation:

  1. Calculate the pooled standard deviation sps_p: sp=(121)×(7.071)2+(131)×(7.351)212+132s_p = \sqrt{\frac{(12 - 1) \times (7.071)^2 + (13 - 1) \times (7.351)^2}{12 + 13 - 2}} sp=11×50.001+12×54.02523=550.011+648.323=1198.31123=52.0977.218s_p = \sqrt{\frac{11 \times 50.001 + 12 \times 54.025}{23}} = \sqrt{\frac{550.011 + 648.3}{23}} = \sqrt{\frac{1198.311}{23}} = \sqrt{52.097} \approx 7.218

  2. Compute the t-statistic: t=40.40537.7577.218×112+113t = \frac{40.405 - 37.757}{7.218 \times \sqrt{\frac{1}{12} + \frac{1}{13}}} t=2.6487.218×0.0833+0.0769=2.6487.218×0.1602=2.6487.218×0.4002=2.6482.8870.917t = \frac{2.648}{7.218 \times \sqrt{0.0833 + 0.0769}} = \frac{2.648}{7.218 \times \sqrt{0.1602}} = \frac{2.648}{7.218 \times 0.4002} = \frac{2.648}{2.887} \approx 0.917

Thus, the t-statistic is approximately 0.917.

Answer: t=0.917t = 0.917.

Let me know if you'd like more details or further clarification on any steps.


Here are 5 related questions that build upon this content:

  1. What are the assumptions required for the two-sample t-test to be valid?
  2. How would the test change if the population variances were unequal?
  3. How do you interpret the p-value for a t-test in hypothesis testing?
  4. What happens if the sample sizes are different but the variances are still assumed to be equal?
  5. How does a paired t-test differ from an independent t-test in terms of design?

Tip: Always check the assumptions (normality, equal variances) before performing a t-test, as these affect the validity of your results.

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Math Problem Analysis

Mathematical Concepts

Two-sample t-test
Hypothesis testing
Statistics

Formulas

t = (X̄1 - X̄2) / (sp * sqrt(1/n1 + 1/n2))
sp = sqrt(((n1 - 1)s1^2 + (n2 - 1)s2^2) / (n1 + n2 - 2))

Theorems

Central Limit Theorem
Two-sample t-test for means

Suitable Grade Level

College/University