Salary information for recent computer science (CS) grads is skewed right because there are a few grads who make a lot more money than others. The population parameters are listed in the table below. If a sample of 10 recent CS grads is selected over and over and the sample mean calculated, what would the shape of the sampling distribution be?

Min

56,000

Q1

76,000

Median

82,000

Mean

91,000

SD

10,250

Q3

92,000

Max

175,000

Question

Salary information for recent computer science (CS) grads is skewed right because there are a few grads who make a lot more money than others.  The population parameters are listed in the table below. If a sample of 10 recent CS grads is selected over and over and the sample mean calculated, what would the shape of the sampling distribution be?

Min

56,000

Q1

76,000

Median

82,000

Mean

91,000

SD

10,250

Q3

92,000

Max

175,000

BlackTom AI · Accepted Answer

Central Limit Theorem (CLT) and sample-size effects determine the shape of a sampling distribution. For sample means: (1) the mean of the sampling distribution equals the population mean, and (2) the standard error equals the population SD divided by sqrt(n). CLT adds that as n gets large, the sampling distribution becomes approximately normal regardless of the population shape; for small n, the sampling distribution will resemble the population shape.

1. Relevant knowledge points and concepts
- Population is skewed right (long right tail).
- Sampling distribution mean = population mean (µ = 91,000).
- Standard error (SE) = σ / sqrt(n) where σ = 10,250 and n = 10.
- CLT: n ≥ ~30 is a common rule of thumb for approximate normality when the population is non-normal.

2. Step-by-step reasoning
- Compute the SE to quantify spread of sample means.
- Evaluate whether n = 10 is large enough for the CLT to force normality.

3. Calculations (intermediate steps)
- sqrt(10) ≈ 3.1623
- SE = 10,250 / 3.1623 ≈ 3,242.6

So the sampling distribution of the sample mean has mean 91,000 and SE ≈ 3,243.

4. Why the correct answer is correct
Because the original population is notably skewed right and the sample size n = 10 is small, the CLT does not guarantee an approximately normal sampling distribution. Averaging reduces but does not eliminate skewness for small samples; therefore the distribution of sample means will still show right skew. The computed SE shows reduced spread around 91,000 but does not change the skew direction.

5. Why other options are incorrect (brief)
- Normal: incorrect because n = 10 is too small for the CLT to override a clearly skewed population.
- Skewed left: incorrect because averaging values from a right-skewed population cannot produce a left skew.
- Uniform: incorrect because there is no basis for a flat, equal-probability distribution of sample means.

6. Conclusion
Given a right-skewed population and n = 10 (small), the sampling distribution of the sample mean will be skewed right. This matches the provided correct answer: Skewed right.

类似问题

Suppose that Quackers now runs 200 people in each experiment. Which of the panels A-E (as shown in Q29 above) most accurately captures what you would expect the sampling distribution of the mean to look like?

Suppose that Quackers only runs 1 person in each experiment. Which of the panels A-E (as shown in Q29 above) most accurately captures what you would expect the sampling distribution of the mean to look like?

What is a sampling distribution?

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