sampling distributions and confidence intervals (for mean)

What 3 items affect the margin of error of a confidence interval?

1. Standard deviation of the population = 𝜎𝑋 2. Sample size = n 3. Z-value

A confidence interval provides a range of ________ values for a population parameter (such as the population mean

Likely

As n increases, what happens to the margin of error (MOE) of your confidence interval? a. Increases b. Decreases c. Stays the same

b. Decreases

As n increases, which of the following statements is true? a. The population standard deviation decreases. b. The standard error of the sample mean decreases. c. The population standard deviation increases. d. The standard error of the sample mean increase

b. The standard error of the sample mean decreases.

Which of the following describes the relationship between parameters and statistics? a. We use parameters to estimate or test statistics b. We use statistics to estimate or test parameters c. Statistics and parameters are the same thing

b. We use statistics to estimate or test parameters

How large does n generally have to be in order for the Central Limit Theorem to take effect? (Assume X does not have a normal distribution.) a. np at least 10 and n(1-p) at least 10 b. n > 30 c. n can be any value d. np ≥ 10

b. n > 30

Let 𝑋 be the sample mean from X's distribution. Suppose X does not have a normaldistribution. The distribution (shape) of 𝑋bar is: a. Exactly normal b. Approximately normal for any sample size by the Central Limit Theorem c. Approximately normal for large enough samples by the Central Limit Theorem d. None of the above

c. Approximately normal for large enough samples by the Central Limit Theorem

What is the symbol we use to represent the mean of the random variable X-bar? a. Mu with subscript X b. Capital X-bar c. Mu with subscript X-bar

c. Mu with subscript X-bar

The set of all possible sample means from all possible samples of size n from the population is known as the: a. Sample mean of the population of X b. Population probability distribution of X c. Sampling distribution of X-bar d. Not enough information to tel

c. Sampling distribution of X-bar

Which of the following is true about the mean of the random variable X-bar as n increases? a. Not enough information to tell b. The mean of X-bar decreases c. The mean of X-bar stays the same d. The mean of X-bar increases

c. The mean of X-bar stays the same

The Central Limit theorem tells us important results that pertain to: a. The mean of the distribution of X-bar b. All of the above c. The shape (type) of the distribution of X-bar d. The standard error of the distribution of X-bar

c. The shape (type) of the distribution of X-bar

If X does not have a normal distribution, which of the following is true about the shape (distribution) of the random variable X-bar? a. X-bar is exactly normal if n > 30 b. X-bar is approximately normal for any n c. X-bar is approximately normal if n > 30 d. X-bar is exactly normal for any n

c. X-bar is approximately normal if n > 30

t/f: The margin of error increases if the sample size increases.

false

t/f: We are 100% confident that our SAMPLE mean is in every confidence interval we make.

false

If X has a normal distribution, which of the following is true about the shape (distribution) of the random variable X-bar? a. X-bar is approximately normal for any n b. X-bar is exactly normal if n > 30 c. X-bar is exactly normal for any n d. X-bar is approximately normal if n > 30

c. X-bar is exactly normal for any n

Bob makes a 95% confidence interval with a random sample of 100 and his margin of error is plus or minus 18. In order to cut the margin of error in half next time, what should Bob's sample size be? (Hint: You CAN determine the answer without knowing the standard deviation.) a. 400 b. 10,000 c. 50 d. 200

a. 400

The Central Limit Theorem is important in statistics because: a. For n large, it says the distribution of the sample mean is approximately normal,regardless of the shape of the population. b. For any sample size, it says the sampling distribution of the sample mean is approximately normal c. For n large, it says the distribution of the sample mean is exactly normal, regardlessof the shape of the population. d. For any sample size, it says the sampling distribution of the sample mean is exactlynormal

a. For n large, it says the distribution of the sample mean is approximately normal,regardless of the shape of the population.

Suppose a pizza place claims it's average pizza delivery time is 10 minutes. Bob makes a 95% confidence interval for the average pizza delivery time for this pizza place and gets (15, 20). What does Bob think about the pizza place's claim? a. He disagrees with the claim. b. He can't disagree with the claim.

a. He disagrees with the claim.

As the confidence level of your confidence interval increases, what happens to the value of Z*? a. Increases b. Decreases c. Stays the sam

a. Increases

As the standard deviation of the population increases, what happens to the MOE? a. Increases b. Decreases c. Stays the same

a. Increases

Which of the following is true about the standard error of the random variable X-bar as n increases? a. The standard error of X-bar decreases. b. The standard error of X-bar increases. c. Not enough information to tell. d. The standard error of X-bar stays the same.

a. The standard error of X-bar decreases.

A confidence interval for the mean equals 𝑥 plus or minus "something". The "something" is called the _________________

margin of error

(t/f) A confidence interval for the mean is known as a range of likely values for the population mean.

true

(t/f) The margin of error of a confidence interval gets larger if the confidence level increases (assume all else stays the same.)

true

t/f: If X has a normal distribution, then 𝑋 has a normal distribution no matter what

true

t/f: Suppose Bob makes a 95% confidence interval and his result is (70, 90). True or false: We can tell what Bob's sample mean must have been.

true

t/f: The margin of error is larger if the standard deviation of the population increases (assume all else stays the same.)

true

If your confidence interval is 95%, what is the value of Z that goes into the confidence interval formula? a. 1.96 b. 2.58 c. 1.645 d. None of the other choices is correct

a. 1.96

Suppose X has a normal distribution and 𝑋bar represents the average of a sample of size n. Which of the following is true? a. 𝑋bar has an approximate normal distribution for any value of n. b. 𝑋bar has an approximate normal distribution if n is large enough. c. 𝑿bar has an exact normal distribution for any value of n. d. 𝑋bar has an exact normal distribution if the sample size is large enoug

c. 𝑿bar has an exact normal distribution for any value of n

Bob wants a 95% confidence interval for the population mean. Previous research shows the population standard deviation is 8. He wants his margin of error to be no more than 4. What should his sample size be? a. 15 b. None of the other choices is correct c. 4 d. 16

d. 16

If X has a normal distribution, could you find a probability for 𝑋bar if n was only 10? a. Yes, because X has a normal distribution already. b. No, because you always need n to be at least 3

a. Yes, because X has a normal distribution already.

A company has developed a new battery, but the average lifetime of all of the batteries it makes is unknown. In order to estimate this average, a sample of 500 batteries is tested and the average lifetime of this sample is found to be 225 hours. The 225 hours is the value of a: a. statistic b. sample c. parameter d. population

a. statistic

The mean of 𝑋bar equals the mean of X when? a. Only if X has a normal distribution b. Only if n is large c. Both a and b are needed d. This statement is always true no matter wha

d. This statement is always true no matter wha

Let 𝑋 be the sample mean from X's distribution. The standard error of 𝑋 is: e. The same as the standard deviation of X f. Greater than the standard deviation of X g. Less than the standard deviation of X

g. Less than the standard deviation of X