The Central Limit Theorem

According to the introduction of Chapter 7, what is the most important conclusion of the Central Limit Theorem regarding the distribution of sample means?

What is the term for the standard deviation of the distribution of the sample means?

According to the Central Limit Theorem, what is the mean of the theoretical sampling distribution of the means (μx̄)?

How does increasing the sample size (n) affect the standard deviation of the sampling distribution of means?

When is it appropriate to use the Finite Population Correction Factor?

A population has a mean of 48 and a standard deviation of 5. If it is sampled repeatedly with samples of size 36, what is the mean and standard deviation of the sample means?

The Central Limit Theorem is described as a 'theorem'. What does this imply?

A researcher is studying a population with a very skewed distribution. What does the Central Limit Theorem say about the sampling distribution of the means if the sample size is large enough?

For the Central Limit Theorem for Proportions, the underlying distribution of the random variable X (number of successes) is which of the following?

A population has a mean of 90 and a standard deviation of 6. If it is sampled repeatedly with samples of size 64, what are the mean and standard deviation of the sample means?

What is the primary reason that one samples a population instead of measuring the entire population?

In the context of the Central Limit Theorem, what does the Law of Large Numbers state?

A fishing boat has 1,000 fish on board. The average weight is 120 pounds with a standard deviation of 6.0 pounds. If a sample of 50 fish is taken, what is the standard error of the mean?

In a town, 37 percent of people have a specific characteristic. If random samples of 30 people are repeatedly taken, what is the expected value of the mean of the sampling distribution of sample proportions?

For the CLT for proportions to apply, and for the binomial distribution to be approximated by the normal distribution, the sample size n must be large enough such that:

What does the standardizing formula Z = (x̄ - μ) / (σ/√n) actually compute?

A company has 800 employees with an average number of workdays between absence for illness of 123 and a standard deviation of 14 days. A sample of 50 employees is examined. Is the Finite Population Correction Factor needed for this problem?

Figure 7.8 on page 322 shows three sampling distributions with sample sizes of 10, 30, and 50. What is the primary visual difference between the distribution for n=10 and the distribution for n=50?

A game is played repeatedly where a player wins one-fifth of the time. If samples of 40 games are taken, what is the standard deviation of the sampling distribution of sample proportions?

The Central Limit Theorem for Proportions is built upon the idea that the binomial distribution can be approximated by which other distribution?

Why is the sampling distribution of the means considered a 'theoretical distribution'?

A population has a mean of 25 and a standard deviation of 2. If it is sampled repeatedly with samples of size 49, what is the mean and standard deviation of the sample means?

What is the key takeaway from Figure 7.9 on page 323, which shows a narrow and a wide sampling distribution?

A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds, with a uniform distribution. What is the mean (μ) and standard deviation (σ) for the weights of one 25-pound lifting weight?

Using the information from the previous question (μ = 25, σ = 0.5774), what is the distribution for the mean weight of 100 25-pound lifting weights?

What is the key difference between the formula for the standard error of the mean when using the Finite Population Correction Factor versus when not using it?

A population has a mean of 17 and a standard deviation of 0.2. If it is sampled repeatedly with samples of size 16, what is the expected value and standard deviation of the sample means?

A virus attacks one in three of the people exposed to it. If samples of 70 people are taken, what is the standard deviation of the sampling distribution of sample proportions?

In the context of Chapter 7, what is the practical significance of the sampling distribution of the means being normally distributed?

In Example 7.2, with 3,000 orders and a sample of 360, the Finite Population Correction Factor is used. Why was it necessary?

The length of time a smartphone battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 smartphones is taken. What is the standard deviation of the individual battery lives?

From the previous question, a sample of 64 smartphones is taken from a population with μ=10 and σ=10. What is the distribution for the mean length of time the 64 batteries last?

The sampling distribution of a parameter is best described as:

A uniform distribution has a minimum of six and a maximum of ten. A sample of 50 is taken. What is the approximate distribution of the sum of the 50 values (Σx)?

The formula for the Finite Population Correction Factor is √((N-n)/(N-1)). Why is N-1 used in the denominator instead of N?

If a population is already normally distributed, what is the minimum sample size needed for the sampling distribution of the means to also be considered normal?

What is the expected value of the sample proportion, E(p')?

An experimental garden has 500 sunflower plants with an average height of 9.3 feet and a standard deviation of 0.5 foot. If a sample of 60 plants is taken, what is the probability the sample will have an average height within 0.1 foot of the true mean?

Which statement best describes the relationship between the Central Limit Theorem and the Law of Large Numbers?

What is the primary trade-off discussed in the text regarding sampling distributions?

A population has a mean of 14 and a standard deviation of 5. If it is sampled repeatedly with samples of size 60, what is the expected value and standard deviation of the sample means?

If you are analyzing a sampling distribution of proportions, and the population proportion 'p' is unknown, what value is commonly used for p to ensure a large enough sample size is calculated?

What is the conclusion of the Central Limit Theorem regarding the underlying distribution of the population data?

For a population of 2,000 trucks with an average weight of 20 tons and a standard deviation of 2 tons, a sample of 50 trucks is taken. What is the approximate probability the sample will have an average weight within 0.5 ton of the population mean?

The symbol 'x̄' represents what concept in Chapter 7?

If a sample size is considered 'large enough', what does the Central Limit Theorem allow us to do?

Why is the Central Limit Theorem considered one of the most powerful and useful ideas in all of statistics?

What is the standard deviation of the sampling distribution of proportions if the population proportion is 0.1 and the sample size is 50?

In Figure 7.2 on page 316, the top panel shows a 'squiggly' unknown population distribution. What does the bottom panel, representing the sampling distribution of means, look like?

A college gives a placement test to 5,000 students. On average, 1,213 place into developmental courses. This represents a proportion of 0.2426. If a sample of 50 is taken, what is the probability at most 12 of them will be in developmental courses?