Q: A pollster surveys 100 people about their preference for summer versus winter. If the true probability of preferring summer is 0.5, what is the expected mean and standard deviation of the number of people preferring summer?

Mean: $$100 \times 0.5 = 50$$, SD: $$\sqrt{100 \times 0.5 \times 0.5} = 5$$

Question 1

A company wants to estimate the proportion of Ottawa consumers interested in buying a new electric truck (like the one shown in the image). They randomly poll 200 consumers and find that 75% say yes. Which of the following is the most important consideration before launching the product based on this sample?

Accepted Answer

Whether the sample is representative of the larger population.

Question 2

Which of the following best describes the difference between categorical and numerical data?

Accepted Answer

Categorical data makes sense to talk about proportions, while numerical data makes sense to talk about averages.

Question 3

In the context of election polling, what is the main reason the Literary Digest poll failed in the 1936 US election?

Accepted Answer

The poll used a non-representative sample, leading to selection bias.

Question 4

Suppose you take a Simple Random Sample (SRS) of size $$n$$ from a population with mean $$\mu$$ and standard deviation $$\sigma. $$What is the standard deviation of the sample mean?

Accepted Answer

$$\frac{\sigma}{\sqrt{n}}$$

Question 5

Which statement about the Central Limit Theorem is correct?

Accepted Answer

It states that the sampling distribution of the sample mean is approximately normal if the sample size is large enough, regardless of the population distribution.

Question 6

A pollster surveys 100 people about their preference for summer versus winter. If the true probability of preferring summer is 0.5, what is the expected mean and standard deviation of the number of people preferring summer?

Accepted Answer

Mean: $$100 	imes 0.5 = 50$$, SD: $$\sqrt{100 	imes 0.5 	imes 0.5} = 5$$

Question 7

Which of the following is NOT a requirement for approximating the binomial distribution with a normal distribution?

Accepted Answer

The probability of success $$p$$ must be close to 0.5.

Question 8

If a sample statistic is unbiased, what does this mean?

Accepted Answer

Its long-run average over all samples equals the population parameter.

Question 9

A business wants to estimate the average time students spend studying for an exam. They take two samples: one of 100 students and one of 1000 students. Which sample will likely produce a statistic with lower variability?

Accepted Answer

The sample of 1000 students.

Question 10

Which formula gives the standard deviation of the sample proportion $$\hat{p}$$ for a binomial distribution with parameters $$n$$ and $$p$$?

Accepted Answer

$$\sqrt{\frac{p(1-p)}{n}}$$

Question 11

Which of the following best describes the sampling distribution?

Accepted Answer

It is the distribution of the values taken by a statistic over repeated samples.

Question 12

A call center manager wants to know if the mean wait time is acceptable. Why is it not enough to just know a list of recent wait times?

Accepted Answer

Because the list may not be representative of the overall population.

Question 13

Which scenario best illustrates selection bias?

Accepted Answer

A poll mailed only to magazine subscribers predicts election results for the entire population.

Question 14

If the sample mean is unbiased, where will the sampling distribution of the sample mean be centered?

Accepted Answer

At the population mean.

Question 15

Which of the following is true about the variability of the sample proportion $$\hat{p}$$?

Accepted Answer

It decreases as $$n$$ increases.

Question 16

A histogram of call lengths is highly skewed, but the histogram of the mean length of 80 calls is nearly normal. What statistical principle does this illustrate?

Accepted Answer

Central Limit Theorem

Question 17

Which of the following is a real-world example of a population parameter?

Accepted Answer

The proportion of Canadians who plan to buy a new electric truck.

Question 18

If you take repeated random samples from a population and calculate the sample mean for each, what will the distribution of these sample means look like as the sample size increases?

Accepted Answer

It will become more concentrated around the population mean.

Question 19

Which of the following is NOT a characteristic of a 'good' statistic?

Accepted Answer

High variability

Sampling Distributions, Estimation, and Data Types in Business Statistics

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