Q: A soda bottler claims that only 5% of cans are underfilled. In a random sample of 200 cans, what is the standard error of the sample proportion?

$$\sqrt{\frac{0.05 \times 0.95}{200}}$$

Question 1

Which of the following best describes the concept of 'sampling error' in statistics?

Accepted Answer

It is the natural discrepancy between a sample statistic and the corresponding population parameter.

Question 2

A population has a mean $$\mu = 80$$ and standard deviation $$\sigma = 12. $$What is the expected value of the sample mean $$\bar{y}$$ for samples of size $$n = 9$$?

Accepted Answer

$$\bar{y} = 80$$

Question 3

Which formula correctly calculates the standard deviation of the sampling distribution of the mean when the population standard deviation $$\sigma is $$known?

Accepted Answer

$$\sigma_{\bar{y}} = \frac{\sigma}{\sqrt{n}}$$

Question 4

If a population is not normal but the sample size is large ($$n \geq 30$$), what does the Central Limit Theorem (CLT) state about the sampling distribution of the sample mean?

Accepted Answer

It will be approximately normal regardless of the population shape.

Question 5

A random sample of $$n = 25 is $$drawn from a population with $$\mu = 100$$ and $$\sigma = 20. $$What is the standard error of the sample mean?

Accepted Answer

$$\sigma_{\bar{y}} = \frac{20}{\sqrt{25}} = 4$$

Question 6

Which of the following is NOT an assumption required for the Central Limit Theorem to apply?

Accepted Answer

Population mean must be zero.

Question 7

If the population standard deviation $$\sigma is $$unknown, which distribution is used for the sampling distribution of the mean?

Accepted Answer

Student's t distribution

Question 8

A soda bottler claims that only 5% of cans are underfilled. In a random sample of 200 cans, what is the standard error of the sample proportion?

Accepted Answer

$$\sqrt{\frac{0.05 	imes 0.95}{200}}$$

Question 9

Which condition must be checked before using the normal approximation for the sampling distribution of proportions?

Accepted Answer

Both $$np \geq 10$$ and $$nq \geq 10$$

Question 10

A population is right-skewed with $$\mu = 65$$ and $$\sigma = 20. If $$you sample $$n = 35$$ students from a class of $$N = 220$$, what adjustment must be made to the standard error?

Accepted Answer

Multiply by the finite population correction factor $$\sqrt{\frac{N-n}{N-1}}$$

Question 11

Which of the following statements about the Student's t distribution is TRUE?

Accepted Answer

It has heavier tails than the normal distribution.

Question 12

If the sample size is drawn without replacement and $$n \u003e 10\% of $$the population, which formula should be used for the standard error of the sample mean?

Accepted Answer

$$\sigma_{\bar{y}} = \frac{\sigma}{\sqrt{n}} \sqrt{\frac{N-n}{N-1}}$$

Question 13

A random sample of $$n = 16$$ scores is obtained from a population with $$\mu = 50$$ and $$\sigma = 16. If $$the sample mean is $$\bar{y} = 58$$, what is the z-score corresponding to the sample mean?

Accepted Answer

Cannot determine (shape of population is unknown)

Question 14

Which of the following best explains why the sampling distribution of the mean becomes more normal as sample size increases?

Accepted Answer

Because the Central Limit Theorem states that the distribution of sample means approaches normality as $$n$$ increases.

Question 15

In which scenario would you use the binomial distribution for the sampling distribution of proportions?

Accepted Answer

When the sample size is small and $$p is $$not close to 0 or 1.

Question 16

A manufacturing company produces batches of 500 electronic items. In a random sample of 80 items, what is the probability that less than 15% are defective if in fact 20% are defective? Which formula should be used to calculate the z-score?

Accepted Answer

$$z = \frac{0.15 - 0.20}{\sqrt{(0.20)(0.80)/80} \sqrt{(500-80)/(500-1)}}$$

Question 17

Which of the following is an unbiased estimator of the population mean?

Accepted Answer

Sample mean $$\bar{y}$$

Question 18

If the population is normal with mean $$\mu$$ and unknown standard deviation $$\sigma$$, what is the sampling distribution of $$\bar{y}$$?

Accepted Answer

$$\bar{y} \sim t(\mu_{\bar{y}} = \mu, \sigma_{\bar{y}} = \frac{s}{\sqrt{n}})$$ with degrees of freedom $$n-1$$

Question 19

Which of the following is TRUE regarding the effect of sample size on the standard error?

Accepted Answer

As sample size increases, the standard error decreases.

Chapter 10: Sampling Distributions – Study Notes for Business Statistics

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