Question

the data in the table represent the ages of the winners of an award for the past five years. use the data to answer questions (a) through (e). (e) construct a sampling distribution for the mean by listing the sample means and their corresponding probabilities. give all the unique sample means in ascending order. (type integers or decimals. type n if there is no solution.)

Explanation:

Step1: Identify Unique Sample Means

First, list all given sample means: 35, 42.5, 36, 37.5, 32.5, 28, 27.5, 33.5, 35, 28.5. Now, find unique ones: 27.5, 28, 28.5, 32.5, 33.5, 35, 36, 37.5, 42.5. Wait, no, original samples (let's check the table structure: maybe 5 samples? Wait, the first column has Sample 1 - 5? Wait, the sample means given: first row sample means: 35, 42.5, 36, 37.5, 32.5? Wait, no, the table has Sample (first column: 45.25, 46.40, 46.27, 45.30, 26.40), then their sample means (35, 42.5, 36, 37.5, 32.5), another sample (25.27, 25.30, 25.27, 40.30, 27.30) with means 28, 27.5, 33.5, 35, 28.5. So all sample means: 35, 42.5, 36, 37.5, 32.5, 28, 27.5, 33.5, 35, 28.5. Now unique means: 27.5, 28, 28.5, 32.5, 33.5, 35, 36, 37.5, 42.5. Wait, but the table below has 5 rows (1 - 5) and then 6 - 10? Wait, maybe the number of samples is 10? Wait, no, the problem says "past five years" but the table has more. Wait, maybe it's a sampling distribution for sample means, so we need to count how many times each mean occurs. Let's list all sample means:

From the table:

Sample 1: mean 35

Sample 2: mean 42.5

Sample 3: mean 36

Sample 4: mean 37.5

Sample 5: mean 32.5

Sample 6: mean 28

Sample 7: mean 27.5

Sample 8: mean 33.5

Sample 9: mean 35

Sample 10: mean 28.5

Now, count frequencies:

27.5: 1

28: 1

28.5: 1

32.5: 1

33.5: 1

35: 2 (samples 1 and 9)

36: 1

37.5: 1

42.5: 1

Wait, no, wait the number of samples: let's check the "Sample" column (first column of data) has 5 entries, then another 5? So total 10 samples. So 10 sample means. Now count occurrences:

- 27.5: 1 (sample 7)

- 28: 1 (sample 6)

- 28.5: 1 (sample 10)

- 32.5: 1 (sample 5)

- 33.5: 1 (sample 8)

- 35: 2 (samples 1 and 9)

- 36: 1 (sample 3)

- 37.5: 1 (sample 4)

- 42.5: 1 (sample 2)

Now, sort unique means in ascending order: 27.5, 28, 28.5, 32.5, 33.5, 35, 36, 37.5, 42.5. Wait, but the table below has 10 rows (1 - 10). Wait, maybe I miscounted. Wait, the first set of samples: 5 samples (45.25, 46.40, 46.27, 45.30, 26.40) with means 35, 42.5, 36, 37.5, 32.5. Second set: 5 samples (25.27, 25.30, 25.27, 40.30, 27.30) with means 28, 27.5, 33.5, 35, 28.5. So total 10 samples. So 10 sample means. Now count each mean's frequency:

- 27.5: 1

- 28: 1

- 28.5: 1

- 32.5: 1

- 33.5: 1

- 35: 2 (two samples: first set sample 1, second set sample 4)

- 36: 1 (first set sample 3)

- 37.5: 1 (first set sample 4)

- 42.5: 1 (first set sample 2)

Wait, that's 1+1+1+1+1+2+1+1+1=10, correct. Now, sort the unique means in ascending order: 27.5, 28, 28.5, 32.5, 33.5, 35, 36, 37.5, 42.5. Wait, but the table below has 10 rows (1 - 5 and 6 - 10), so maybe 10 sample means, but some are repeated. Wait, no, the problem says "Give all the unique sample means in ascending order". So first, list unique means:

27.5, 28, 28.5, 32.5, 33.5, 35, 36, 37.5, 42.5. Wait, but let's check the values again. 27.5 is less than 28, then 28.5, 32.5, 33.5, 35, 36, 37.5, 42.5. Yes.

Step2: Calculate Probabilities (Frequency / 10)

Now, for each unique mean, count how many times it occurs (frequency) and divide by 10 (total samples) to get probability.

- 27.5: frequency 1 → probability 1/10 = 0.1

- 28: frequency 1 → 0.1

- 28.5: frequency 1 → 0.1

- 32.5: frequency 1 → 0.1

- 33.5: frequency 1 → 0.1

- 35: frequency 2 → 2/10 = 0.2

- 36: frequency 1 → 0.1

- 37.5: frequency 1 → 0.1

- 42.5: frequency 1 → 0.1

Wait, but the table below has two sections: 1 - 5 and 6 - 10 for Sample Mean. Wait, maybe the initial samples are 10, so 10 sample means. So the unique means in ascending order: 27.5, 28, 28.5, 32.5, 33.5, 35, 36, 37.5, 42.5. But wait, maybe I made a mistake in the sample means. Let's re-express the sample means:

First group (5 samples):

Sample 1: mean 35

Sample 2: mean 42.5

Sample 3: mean 36

Sample 4: mean 37.5

Sample 5: mean 32.5

Second group (5 samples):

Sample 6: mean 28

Sample 7: mean 27.5

Sample 8: mean 33.5

Sample 9: mean 35

Sample 10: mean 28.5

Yes, so 10 means. Now unique means sorted: 27.5, 28, 28.5, 32.5, 33.5, 35, 36, 37.5, 42.5. Wait, but 9 unique means? But the table below has 10 rows (1 - 10), so maybe I missed a mean? Wait, no, 10 means with 35 repeated. So the unique means are 9, but the table has 10 rows? Wait, maybe the problem's table has 10 samples, so 10 sample means, some repeated. So the sampling distribution table will have each unique mean and its probability (frequency/10).

Now, let's list them in ascending order:

1. Sample Mean: 27.5, Probability: 1/10 = 0.1

2. Sample Mean: 28, Probability: 1/10 = 0.1

3. Sample Mean: 28.5, Probability: 1/10 = 0.1

4. Sample Mean: 32.5, Probability: 1/10 = 0.1

5. Sample Mean: 33.5, Probability: 1/10 = 0.1

6. Sample Mean: 35, Probability: 2/10 = 0.2

7. Sample Mean: 36, Probability: 1/10 = 0.1

8. Sample Mean: 37.5, Probability: 1/10 = 0.1

9. Sample Mean: 42.5, Probability: 1/10 = 0.1

Wait, but the table in the problem has two parts: first 5 (1 - 5) and next 5 (6 - 10) for Sample Mean and Probability. Maybe the number of unique means is 10? No, because 35 is repeated. So perhaps the initial data has 10 samples, so 10 sample means, with 35 appearing twice. So the unique means are 9, but the table has 10 rows? Maybe a typo, but proceed with the unique means and their probabilities.

Answer:

Unique sample means in ascending order: 27.5, 28, 28.5, 32.5, 33.5, 35, 36, 37.5, 42.5
Corresponding probabilities: 0.1, 0.1, 0.1, 0.1, 0.1, 0.2, 0.1, 0.1, 0.1

(To fill the table:
Row 1: Sample Mean 27.5, Probability 0.1
Row 2: Sample Mean 28, Probability 0.1
Row 3: Sample Mean 28.5, Probability 0.1
Row 4: Sample Mean 32.5, Probability 0.1
Row 5: Sample Mean 33.5, Probability 0.1
Row 6: Sample Mean 35, Probability 0.2
Row 7: Sample Mean 36, Probability 0.1
Row 8: Sample Mean 37.5, Probability 0.1
Row 9: Sample Mean 42.5, Probability 0.1
Row 10: Wait, no, maybe I miscounted. Wait, 10 samples, so 10 means. Let's list all 10 means: 35, 42.5, 36, 37.5, 32.5, 28, 27.5, 33.5, 35, 28.5. Now sort them: 27.5, 28, 28.5, 32.5, 33.5, 35, 35, 36, 37.5, 42.5. Ah! Here's the mistake: I forgot that 35 appears twice, so in the sorted list, it's 27.5, 28, 28.5, 32.5, 33.5, 35, 35, 36, 37.5, 42.5. So unique means with their counts:

• 27.5: 1

• 28: 1

• 28.5: 1

• 32.5: 1

• 33.5: 1

• 35: 2

• 36: 1

• 37.5: 1

• 42.5: 1

Wait, that's 1+1+1+1+1+2+1+1+1=10. So when sorted, the 10 means are: 27.5, 28, 28.5, 32.5, 33.5, 35, 35, 36, 37.5, 42.5. But the problem says "unique sample means", so we take each unique value once, then their probabilities (frequency/10). So the unique means are 27.5, 28, 28.5, 32.5, 33.5, 35, 36, 37.5, 42.5 (9 unique), but the table has 10 rows. Maybe the problem has 10 samples, so 10 means, some repeated, and we list all means (including repeats) in ascending order? No, the problem says "unique sample means". So correct sorted unique means: 27.5, 28, 28.5, 32.5, 33.5, 35, 36, 37.5, 42.5. Probabilities: 1/10, 1/10, 1/10, 1/10, 1/10, 2/10, 1/10, 1/10, 1/10.

So the sampling distribution table (unique means and probabilities) is:

Sample Mean	Probability
28	0.1
28.5	0.1
32.5	0.1
33.5	0.1
35	0.2
36	0.1
37.5	0.1
42.5	0.1

(Note: If the table requires 10 rows, maybe there's a mistake, but based on unique means, this is the sampling distribution.)