Question

a large company has two major departments, development and marketing. twelve employees are randomly selected from each department, and the age of each employee, in years, is recorded in the accompanying samples. both departments have an employee who is 56 years old. in which department is it more unusual to have a 56-year-old employee? use a ti-83, ti-83 plus, or ti-84 calculator to calculate the z - scores. round to two decimal places. do not round until youve calculated your final answer development marketing 49 56 51 22 23 24 28 32 56 25 44 29 47 48 38 30 33 29 provide your answer below: development z - score: marketing z - score:

Explanation:

Step1: Calculate mean and standard deviation for Development department

First, we list the ages in the Development department: 49, 51, 23, 28, 56, 44, 47, 38, 33. Wait, the problem says twelve employees are selected from each department, but the table shows 9 entries? Wait, maybe it's a typo, but let's use the given data. Wait, no, maybe the table is incomplete, but the problem says "twelve employees are randomly selected from each department" but the table has 9? Wait, maybe it's a mistake, but let's proceed with the given data. Wait, no, maybe I miscounted. Let's check: Development column: 49, 51, 23, 28, 56, 44, 47, 38, 33. That's 9 values. Marketing: 56, 22, 24, 32, 25, 29, 48, 30, 29. Also 9. Maybe the problem has a typo, but we'll use these 9 values.

First, calculate the mean (\(\mu\)) and standard deviation (\(\sigma\)) for Development.

Sum of Development ages: \(49 + 51 + 23 + 28 + 56 + 44 + 47 + 38 + 33\)

Let's calculate:

49 + 51 = 100

100 + 23 = 123

123 + 28 = 151

151 + 56 = 207

207 + 44 = 251

251 + 47 = 298

298 + 38 = 336

336 + 33 = 369

Mean (\(\mu_{D}\)) = \(369 / 9 = 41\)

Now, calculate the squared deviations from the mean:

For 49: \((49 - 41)^2 = 8^2 = 64\)

51: \((51 - 41)^2 = 10^2 = 100\)

23: \((23 - 41)^2 = (-18)^2 = 324\)

28: \((28 - 41)^2 = (-13)^2 = 169\)

56: \((56 - 41)^2 = 15^2 = 225\)

44: \((44 - 41)^2 = 3^2 = 9\)

47: \((47 - 41)^2 = 6^2 = 36\)

38: \((38 - 41)^2 = (-3)^2 = 9\)

33: \((33 - 41)^2 = (-8)^2 = 64\)

Sum of squared deviations: \(64 + 100 + 324 + 169 + 225 + 9 + 36 + 9 + 64\)

Calculate:

64 + 100 = 164

164 + 324 = 488

488 + 169 = 657

657 + 225 = 882

882 + 9 = 891

891 + 36 = 927

927 + 9 = 936

936 + 64 = 1000

Variance (\(s^2\)) = \(1000 / (9 - 1) = 1000 / 8 = 125\) (wait, no: for sample standard deviation, we use \(n - 1\), but if it's population, \(n\). But the problem says "employees" so maybe population? Wait, the company has two departments, and we selected 12 employees from each, but the table has 9. Maybe it's a sample. Wait, the problem says "the accompanying samples", so sample. So we use \(n - 1\) for sample standard deviation.

Wait, but the problem says "use a TI-83, TI-83 Plus, or TI-84 calculator", so maybe we should use the calculator steps. But since we're doing manually, let's check again.

Wait, maybe I made a mistake in the number of values. Wait, the problem says "twelve employees are randomly selected from each department", but the table shows 9. Maybe it's a typo, but let's proceed with the given data. Alternatively, maybe the table is correct and the "twelve" is a mistake. Let's assume the table has 9 values for each.

So for Development:

Mean (\(\mu_D\)) = 369 / 9 = 41

Sample standard deviation (\(s_D\)): \(\sqrt{1000 / 8} = \sqrt{125} \approx 11.18\)

Now, the z-score for 56 in Development: \(z_D = (56 - 41) / 11.18 \approx 15 / 11.18 \approx 1.34\)

Now for Marketing department:

Ages: 56, 22, 24, 32, 25, 29, 48, 30, 29

Sum of Marketing ages: \(56 + 22 + 24 + 32 + 25 + 29 + 48 + 30 + 29\)

Calculate:

56 + 22 = 78

78 + 24 = 102

102 + 32 = 134

134 + 25 = 159

159 + 29 = 188

188 + 48 = 236

236 + 30 = 266

266 + 29 = 295

Mean (\(\mu_M\)) = 295 / 9 ≈ 32.78

Squared deviations from the mean:

56: \((56 - 32.78)^2 = (23.22)^2 ≈ 539.17\)

22: \((22 - 32.78)^2 = (-10.78)^2 ≈ 116.21\)

24: \((24 - 32.78)^2 = (-8.78)^2 ≈ 77.09\)

32: \((32 - 32.78)^2 = (-0.78)^2 ≈ 0.61\)

25: \((25 - 32.78)^2 = (-7.78)^2 ≈ 60.53\)

29: \((29 - 32.78)^2 = (-3.78)^2 ≈ 14.29\)

48: \((48 - 32.78)^2 = (15.22)^2 ≈ 231.65\)

30: \((30 - 32.78)^2 = (-2.78)^2 ≈ 7.73\)

29: \((29 - 32.78)^2 = (-3.78)^2 ≈ 14.29\)

Sum of squared deviations: 539.17 + 116.21 + 77.09 + 0.61 + 60.53 + 14.29 + 231.65 + 7.73 + 14.29

Calculate:

539.17 + 116.21 = 655.38

655.38 + 77.09 = 732.47

732.47 + 0.61 = 733.08

733.08 + 60.53 = 793.61

793.61 + 14.29 = 807.9

807.9 + 231.65 = 1039.55

1039.55 + 7.73 = 1047.28

1047.28 + 14.29 = 1061.57

Sample variance (\(s_M^2\)) = 1061.57 / 8 ≈ 132.696

Sample standard deviation (\(s_M\)) = \(\sqrt{132.696} ≈ 11.52\)

Now, z-score for 56 in Marketing: \(z_M = (56 - 32.78) / 11.52 ≈ 23.22 / 11.52 ≈ 2.01\)

Wait, but this contradicts the earlier thought. Wait, maybe I made a mistake in the number of values. Wait, the problem says "twelve employees", so maybe the table is missing 3 values. But since we have to use the given data, maybe the original problem has 9 values. Alternatively, maybe I should use the calculator method.

Alternatively, let's use the TI-83 steps:

For Development:

Enter the data into L1: 49, 51, 23, 28, 56, 44, 47, 38, 33

Calculate 1-Var Stats:

Mean (\(\bar{x}\)) = 41, Sample standard deviation (Sx) = \(\sqrt{125} ≈ 11.1803\)

Z-score for 56: (56 - 41) / 11.1803 ≈ 15 / 11.1803 ≈ 1.34

For Marketing:

Enter data into L2: 56, 22, 24, 32, 25, 29, 48, 30, 29

1-Var Stats:

Mean (\(\bar{x}\)) = 295 / 9 ≈ 32.7778, Sample standard deviation (Sx) = \(\sqrt{1061.57 / 8} ≈ \sqrt{132.696} ≈ 11.52\)

Z-score: (56 - 32.7778) / 11.52 ≈ 23.2222 / 11.52 ≈ 2.01

Wait, but the z-score for Marketing is higher, meaning it's more unusual. But wait, maybe the number of values is 12. Let's check again. Maybe the table is missing 3 values. Let's assume that the table has 9 values, but the problem says 12. This is a problem. Alternatively, maybe the original data has 12 values, and the table is a typo. But since we have to proceed, let's check with the given data.

Wait, maybe I made a mistake in the mean for Development. Let's recalculate the sum:

Development: 49, 51, 23, 28, 56, 44, 47, 38, 33

49 + 51 = 100

100 + 23 = 123

123 + 28 = 151

151 + 56 = 207

207 + 44 = 251

251 + 47 = 298

298 + 38 = 336

336 + 33 = 369. 369 / 9 = 41. Correct.

Marketing: 56, 22, 24, 32, 25, 29, 48, 30, 29

56 + 22 = 78

78 + 24 = 102

102 + 32 = 134

134 + 25 = 159

159 + 29 = 188

188 + 48 = 236

236 + 30 = 266

266 + 29 = 295. 295 / 9 ≈ 32.7778. Correct.

Now, z-score for Development: (56 - 41) / (standard deviation). Wait, if it's population standard deviation (since we selected all employees? No, it's a sample). Wait, the problem says "the age of each employee, in years, is recorded in the accompanying samples", so sample. So we use sample standard deviation.

For Development, sample standard deviation: \(\sqrt{1000 / 8} = \sqrt{125} ≈ 11.18\)

Z-score: (56 - 41) / 11.18 ≈ 1.34

For Marketing, sample standard deviation: \(\sqrt{1061.57 / 8} ≈ 11.52\)

Z-score: (56 - 32.78) / 11.52 ≈ 2.01

Since the z-score for Marketing is higher (2.01 > 1.34), it's more unusual to have a 56-year-old in Marketing.

But wait, maybe I made a mistake in the number of values. Let's check the problem again. The problem says "twelve employees are randomly selected from each department", but the table shows 9. Maybe it's a typo, and the table has 9. Alternatively, maybe the original data has 12. Let's assume that the table is correct and the "twelve" is a mistake. So proceeding with the given data.

Answer:

Development z-score: \(\boxed{1.34}\)

Marketing z-score: \(\boxed{2.01}\)

(And the more unusual department is Marketing, since its z-score is higher.)