Stats

The area to the right of 0 in a standard normal distribution is: a. equal to .5 minus the area to the left of 0 b. .5 c. equal to the area to the right of 0 in any normal distribution d. undetermined unless μ and σ are known e. 1

B (Cannot be A because μ is 0 and therefore it is centered at 0. This would make it unsymmetrical because the left would be .5 and the right would be 0. Cannot be C because the normal distribution does not have the same parameters as a standard normal distribution. Cannot be D because standard normal distribution parameters are known. Cannot be E because it is symmetrical about 0, meaning that you take the standard deviation and divide it by 2, so that would be 1/2=0.5. It is B because is symmetrical on both sides.)

P (-2 < Z < 0) a. 0.0000 b. 0.4772 c. -0.4772 d. 0.0228 e. -0.0228

B (Probabilities must be between 0 and 1, therefore it cancels out C and E. To solve, look up the values and subtract the value of 0 from -2. It should be 0.5000-0.228, which is B)

Suppose for a sample of size 75 we have a mean of 13 and a standard deviation of 2. Then a 96% confidence interval for the mean of the sampled population would have left and right ends points 13-(2.06)(2/square root of 75) and 13+(2.06)(2/square root 75) respectively. If we constructed such a confidence interval for 100 samples, we could expect about _____ of them to contain the population mean. a. 4 b. 2 c. 96 d. 2.58 e. 75

C (Multiply 100 by 96%)

The area under a probability density function of a continuous random variable correspond to probabilities for the random variable X. That implies that: a. X is a discrete random variable b. X is a normal random variable c. the total area under a probability density function equals 1. d. the probability distribution is always mound-shaped. e. all of the above

C (a discrete random variable has possible values that can be listed - generally has only a finite number of possible values. Cannot be A because random variable X has values that correspond to all the values under the X, which are infinite. Cannot be B because a random variable has values that depend on chance. Cannot be D because not all continues random variables have mound shaped density functions. It is C because the area beneath the density function always equals 1.)

The standard normal distribution: a. is skewed b. is a discrete probability distribution c. always has a mean equal to 0 and standard deviation equal to 1 d. has a mean μ, equal to 1 and a variance σ2, equal to 0 e. is symmetric about σ2

C (cannot be A because it is not skewed. A standard normal distribution is mound shaped. Cannot be b because it does not describe a discrete random variable. Cannot be d because the mean is equal to 0 and the standard deviation is 1, not the other way around. Cannot be E because it is symmetric about the mean, not the standard deviation. It is C because the mean is always equal to 0 and the standard deviation is always 1)

It is known that the distribution of X is normal with mean equal to 10 and a standard deviation of 4. Find the probability that X is greater than 16. a. 0.4332 b. 0.4987 c. 0.2013 d. 0.9332 e. 0.0668

D

P (Z > -0.07) a. 0.4721 b. 0.0279 c. 0.2580 d. 0.5279 e. 0.7580

E (You find the value of -0.7 and then subtract that from 1 because Z is greater than that value. Therefore it is 1-0.2420, which is E)

An estimator is said to be consistent if the estimator gets closer to the parameter it is estimating as the population variance increases.

False (As variance increases, a consistent estimator should get closer to the desired value)

According to the Central Limit Theorem the standard error of X bar decreases as the population increases

False (As σ/standard error decreases, the variance decreases)

The Central Limit Theorem is not applicable to samples taken from non-normaly distributed populations

False (CLT can be applied to everything)

A disadvantage of using a 99 percent confidence level rather than a 95 percent confidence level for parameter estimation is that larger samples must be taken for establishing higher levels of confidence.

False (Confidence level has nothing to do with sample size)

All students' T based confidence intervals for a population mean mu will be of equal width provided the level of confidence is the same for each interval

False (Different sample sizes will require different t-based confidence intervals)

Every estimator has a sampling distribution having parameters that are totally unrelated to the population of interest from which the data was sampled.

False (Estimators are used to guess parameters, and therefore are related)

Increasing the confidence level decreases the width of the interval estimate for the population mean, given that the sample data remains the same.

False (If s/square root n remains the same, then increasing the confidence level will increase the range one goes out to find the mean, therefore increasing the width.)

A point estimate is preferred over an interval estimate from a statistical point of view.

False (Interval estimates give a wider range where the mean could fall, which is more accurate)

Consider a population with a population mean μ and a standard deviation σ. The Central Limit Theorem states that for a sufficiently large sample size, the sampling distribution of the sample mean, x̄, is approximately a normal distribution with expected value (mean) μ and standard deviation σ

False (It is not σ, rather it is s2/square root sample size)

The Central Limit Theorem is important because, provided the sample size is sufficiently large, it can be applied for determining the sampling distribution of the sample median without assuming the distribution of the population of interest is known.

False (It is used to determine the sample mean, not the sample median)

The Central Limit Theorem states that for a sufficiently large sample size the sampling distribution of the sample mean, X Bar, is N(μ,σ) where μ is the mean of the population of interest, σ is the standard deviation of the population of interest, and n is the sample size.

False (It is σ/square root of n, not σ)

The central limit theorem plays a key role in developing a confidence interval for the population mean μ provided the sample size is not too large.

False (Needs large size)

A 95 percent confidence interval has a probability of .95 that the parameter μ will be contained in the confidence interval

False (No probability is associated with μ because it is the center)

The Central Limit Theorem yields the approximate sampling distribution of the population mean provided the sample size is sufficiently large.

False (No such thing: sampling distribution of a parameter does not exist)

According to the Central Limit theorem, the samplilng distribution of the sample mean for sufficiently large sample sizes will be identical to the distribution of the sampled (parent) population

False (Regardless of the original shape, X bar will be normal with mean μ and sigma/square root n)

According to the Central Limit Theorem, the sampling distribution of the sample mean for sufficiently large sample sizes will be similar to the probability density function of the population of interest.

False (Regardless of what the density function looks like, x bar will be approximately normal and form a mound shape)

Confidence intervals give no information regarding the precision of an estimator

False (Smaller confidence intervals give more information than larger confidence intervals)

A normal distribution always has an expected value of zero.

False (Standard normal distributions always have an expected value of zero)

Parameters are used to estimate statistics.

False (Statistics are used to estimate parameters)

The width of a confidence interval for μ depends only on the parameter value it is attempting to estimate.

False (The parameter value sets where the center is, and does not affect the width)

The Central Limit Theorem states that for sufficiently large sample sizes, i..e, 30 or more, the sampling distribution of the sample mean, X bar, is N (μ, σ/n) where mu is the mean of the parent population, σ is the standard deviation of the parent population, and n is the sample size.

False (must be σ/square root n)

There is only one symmetric bell-shaped density function that has mean zero and standard deviation one.

False (there are many)

Statistical inference permits us to draw conclusions concerning a population and its population parameters based on sample data.

True

The standard error of the sample mean X bar does not depend on the expected value of the population of interest that is being sampled.

True

An estimator, if said to be unbiased, is the expected value of the estimator is exactly the parameter which the estimator is attempting to estimate.

True (A statistic is said to be an unbiased estimate of a given parameter when the mean of the sampling distribution of that statistic can be shown to be equal to the parameter being estimated)

The width of a confidence interval for the population mean mu is dependent on the sample standard deviation.

True (Different sigma and s will have different widths)

Sampling error is a source of error for every estimator.

True (Estimators are guesses based off of samples, and therefore have sampling error as a source of error)

The width of a confidence interval for μ decreases as the sample size increases for fixed confidence level, provided the sample standard deviation remains constant.

True (If CI and S stay the same, the square root of n increasing will decrease the width)

The Central Limit Theorem can be applied without regard to the shape of the population density function provided the sample size is sufficiently large.

True (The CLT allows us to disregard the density function so long as we have a large enough sample pool)

The Central Limit Theorem applies to populations which are modeled as discrete random variables.

True (The Central Limit Theorem can apply to anything as long as it has enough samples)

The expected value of the sample mean is identical to the population mean.

True (The sample mean is an estimator for the population mean, and is expected to be similar)

The expected value of the sample mean, x bar, is the same as the population mean of the sampled population.

True (expected value is identical to μ)

The Central Limit Theorem is important because it can ALWAYS be applied without assuming the distribution of the population of interest, provided the sample size is sufficiently large and the population variance is finite.

True (large enough and no skew present)