Quiz 4 (chapters 7-8)

A random sample of n = 16 scores is obtained from a population with sigma = 12. If the sample mean is 6 points greater than the population mean, what is the z-score for the sample mean? +6.00 +2.00 +1.00 cannot be determined without knowing the population mean

+2.00 (The z-score for the sample mean must be compared to the distribution of sample means, and the standard deviation of the distribution of sample means (aka, the standard error) for samples of n=16 is 12/sqrt(16) = 3. Six is twice three. Hence, a sample mean of six points above mu has a z-score of +2.00.)

Samples of size n=9 are selected from a population with mu = 80 and sigma = 18. What is the standard error for the distribution of sample means? 80 6 18 2

6 (the standard error of the mean -- that is, the standard deviation of the distribution of sample means -- is 6, because 18/sqrt(9) = 6.)

Samples of size n=9 are selected from a population with mu = 80 and sigma = 18. What is the expected value for the distribution of sample means? 26.67 80 18 6

80 (the expected value (of the mean) for the distribution of sample means is 80, the same as the underlying population mean. The sample size is not relevant here.--CENTRAL LIMIT THEOREM)

Which combination of factors will produce the smallest value for the standard error? a large sample and a large standard deviation a small sample and a large standard deviation a small sample and a small standard deviation a large sample and a small standard deviation

a large sample and a small standard deviation (This is specified by the Central Limit Theorem's formula for standard error, which has standard deviation in the numerator and sample size in the denominator.)

Even if a treatment has an effect, it is still possible to obtain a treated sample mean that is very similar to the original (untreated) population mean. What outcome is likely if this happens? experimenter will reject H0 and make a Type I error experimenter will fail to reject H0 and make a Type II error experimenter will correctly fail to reject H0 (i.e., correctly retain H0) experimenter will correctly reject H0

experimenter will fail to reject H0 and make a Type II error

Even if a treatment has no effect, it is still possible to obtain an extreme sample mean -- because of sampling error alone -- that is very different from the original, untreated population mean. What outcome is likely if this happens? experimenter will reject H0 and make a Type I error experimenter will fail to reject H0 and make a Type II error experimenter will correctly reject H0 experimenter will correctly fail to reject H0 (i.e., correctly retain H0)

experimenter will reject H0 and make a Type I error (It is likely that the experimenter will interpret that extremely unlikely sample mean as evidence of a real effect, and consequently reject H0 and make a Type I error. (This is not hugely unusual; by definition, on average, one out of every 20 studies with alpha = 0.05 will make a Type I error).)

Which of the following accurately describes the critical region? outcomes with a high probability whether or not the null hypothesis is true outcomes with a very low probability whether or not the null hypothesis is true outcomes with a very low probability if the null hypothesis is true outcomes with a high probability if the null hypothesis is true

outcomes with a very low probability if the null hypothesis is true (The critical region of a distribution -- the tail or tails -- corresponds to outcomes with a very low probability if the null hypothesis is true. The judgement step of a hypothesis test essentially says that "if this outcome is unusual enough under H0 -- more unusual than our choice of alpha -- then it is too unusual to believe and hence we reject H0.)

Which of the following accurately describes the effect of increasing the sample size n ? reduces the standard error and has no effect on the risk of a Type I error increases the risk of a Type I error and has no effect on the standard error reduces the risk of a Type I error and has no effect on the standard error increases the standard error and has no effect on the risk of a Type I error

reduces the standard error and has no effect on the risk of a Type I error (increasing the sample size reduces the standard error (denominator = square root of n), but has no effect on the risk of a Type I error (which depends solely on the investigator's choice of alpha).)

A random sample of n=36 scores is selected from a population. Given just this information, which of the following distributions definitely will be normal? The scores in the population will form a normal distribution. Neither the sample, nor the population, nor the distribution of sample means will definitely be normal. The scores in the sample will form a normal distribution. The distribution of sample means will form a normal distribution.

the distribution of sample means will form a normal distribution. (Distributions of sample means are reliably normal when the sample size n is greater than 30. (It is possible that the population distribution also is normal, but we don't have that information).)

What term is used to identify the mean of the distribution of sample means? the central limit theorem the sample mean the standard error of M the expected value of M

the expected value of M (the expected value of M is the term for the mean of the distribution of sample means. As the sample size n increases, it becomes a better and better estimate of the population mean μ (which is usually unknown in real life).)

What term is used to identify the standard deviation of the distribution of sample means? the sample mean the standard error of M the expected value of M the central limit theorem

the standard error of M (aka the standard error of the mean, aka the standard error, aka the SEM) is the term for the standard deviation of the distribution of sample means. As the sample size n increases, the SEM becomes smaller and smaller, because higher sample sizes make every individual sample mean a better estimate of the population mean μ, hence a distribution of these means is narrower.

By selecting a larger alpha level (say, 0.05 as opposed to 0.01), a researcher is __________. increasing the risk of a Type I error attempting to make it easier to reject H0 better able to detect a treatment effect if it exists (increase power) All of the other answers are correct.

All of the other answers are correct

The critical boundaries (Zcrit) for a hypothesis test are z = +1.96 and z = -1.96. If the z-score for the sample data is z = -1.90, then what is the correct statistical decision? Fail to reject H0 Reject H0 Reject H1 Fail to reject H1

Fail to reject H0 ( Correct: the z-score does not exceed the critical boundaries (Zcrit) on either side, hence one fails to reject H0. This does not mean that one can reject H1, because this result could arise, for example, from simply having too little evidence -- it does not reliably provide evidence that H1 is wrong, but simply shows that there is not enough evidence to say that H1 is correct (by rejecting H0). Note that simply declaring the Zcrit boundaries to be +1.96 and -1.96 tells you that (1) the test is two-tailed and (2) that the alpha value is 0.05 - specifically, 0.025 at each of the two tails.)

What happens to the expected value of M as sample size increases? It also increases The effect on the expected value as sample size increases is not predictable It stays constant It decreases

It stays constant (The expected value of M is always equal to the population mean, because M is an unbiased statistic.)

For a normal population with mu = 40 and sigma = 10, which of the following samples is LEAST likely to be obtained? M < 42 for a sample of n = 100 M < 44 for a sample of n = 100 M < 44 for a sample of n = 4 M < 42 for a sample of n = 4

M < 42 for a sample of n = 4 (First, given a mean of 40 in a normal distribution, it is always less likely (for any given sample size) to get M < 42 than it is to get M < 44. Second, the mean of a smaller sample is always more likely to be further from the underlying population mean (because the standard error is larger), and this means that it is less likely to be within a given distance of the mean compared to the mean of a larger sample.)

A sample of n=4 scores is obtained from a population with mu = 70 and sigma = 8. If the sample mean corresponds to a z-score of +2.00, then what is the value of the sample mean? M = 78 M = 86 M = 74 M = 72

M = 78 ( The z-scores of sample means must be assessed with respect to the distribution of sample means of the same size; for samples of n=4, the standard error here is sigma/sqrt(n) = 8/sqrt(4) = 4. Hence, a z-score of +2.00 for this sample mean corresponds to 2*4 points above the population mean, i.e. 8 above the mean of 70, i.e. 78.)

A random sample of n=4 scores is selected from a population. Given just this information, which of the following distributions definitely will be normal? Neither the sample, nor the population, nor the distribution of sample means will definitely be normal. The distribution of sample means will form a normal distribution. The scores in the population will form a normal distribution. The scores in the sample will form a normal distribution.

Neither the sample, nor the population, nor the distribution of sample means will definitely be normal. ((it is possible that the population distribution is normal, but we don't have that information). The sample size is too small for the distribution of sample means to be normal (need at least n=30) given that we don't know anything about the population's normality, and there is no way that a sample of just four scores could be considered normally distributed.)

A researcher conducts a hypothesis test to evaluate the effect of a treatment that is expected to increase scores. The hypothesis test produces a z-score of z=+2.37. If the researcher is using a one-tailed test, what is the correct statistical decision? Cannot answer without additional information Fail to reject the null hypothesis with either alpha = 0.05 or alpha = 0.01 Reject the null hypothesis with alpha = 0.05 but not with alpha = 0.01 Reject the null hypothesis with either alpha = 0.05 or alpha = 0.01

Reject the null hypothesis with either alpha = 0.05 or alpha = 0.01 (A z-score of +2.37 or greater corresponds to a probability of p = 0.0089. With a one-tailed test where the entire critical region is at the positive end, p < alpha, so you can reject the null hypothesis. (Obviously, you then can also reject the null hypothesis at higher alpha levels as well). Note that if the test had been two-tailed, then only half of the critical region would have been at the positive end, so the alpha(+) would have been 0.025, and one could not have rejected H0 at this alpha level. (At an alpha level of 0.05, one could reject H0 even using a two-tailed test).)

What happens to the standard error of M as sample size increases? The effect on the standard error as sample size increases is not predictable It stays constant It also increases It decreases

it decreases. (The standard error of M equals the standard deviation of the population divided by the square root of n (Central Limit Theorem), hence it decreases as n increases.)

A researcher administers a treatment to a sample of participants selected from a population with mu = 80. If the researcher obtains a sample mean of M=88, which combination of factors is most likely to result in rejecting the null hypothesis? standard deviation sigma = 10 and alpha = 0.05 standard deviation sigma = 10 and alpha = 0.01 standard deviation sigma = 5 and alpha = 0.05 standard deviation sigma = 5 and alpha = 0.01

standard deviation sigma = 5 and alpha = 0.05 (The combination of factors most likely to result in rejecting H0 is to have the smaller standard deviation (sigma = 5) and the higher alpha level (0.05). Smaller standard deviation makes the two distributions (H0 and H1) more distinct from one another, increasing power and the likelihood of rejecting H0, and the higher alpha level directly increases the probability of rejecting H0 (irrespective of whether H0 should in reality be rejected).)

What is measured by the numerator of the z-score test statistic? [remember, the z-score test statistic is: z = (M-mu)/sigma] whether or not there is a significant difference between M and mu the average distance between M and mu that would be expected if H0 were true the actual distance between M and mu the position of the sample mean relative to the critical region

the actual distance between M and mu ((M-mu) is the actual distance between the treated sample mean M and the untreated population mean mu. This difference, divided by the population SD, is how you convert that treated sample mean M to its corresponding z-score.)

What is measured by the denominator of the z-score test statistic? [remember, the z-score test statistic is: z = (M-mu)/sigma] the position of the sample mean relative to the critical region the average distance between M and mu that would be expected if H0 were true whether or not there is a significant difference between M and mu the actual distance between M and mu

the average distance between M and mu that would be expected if H0 were true (The denominator, sigma, aka the population standard deviation, is the average distance between M and mu that would be expected if H0 were true. Hence, the ratio of the numerator (the actual difference) to the denominator (the expected difference under H0) produces you a value -- the z-score -- that intrinsically tells you how unusual this particular sample mean M is.)

For a population with a mean of mu = 80 and sigma = 20, the distribution of sample means based on n = 16 will have an expected value of _____ and a standard error of ______. 5, 80 20, 20 80, 1.25 80, 5

the distribution of sample means for n=16 will have an expected value of 80 (the same as the population mean mu) and a standard error of 5 (the standard deviation divided by the square root of n, as specified by the Central Limit Theorem).

If a sample of n = 4 scores is obtained from a population with mu = 70 and sigma = 12, then what is the z-score corresponding to a sample mean of M = 76? z = +2.00 z = +0.25 z = +0.50 z = +1.00

z = +1.00 (The z-scores of sample means must be assessed with respect to the distribution of sample means of the same size; for samples of n=4, the standard error here is sigma/sqrt(n) = 12/sqrt(4) = 6. The sample mean obtained is six points above the population mean, which is one standard error above the population mean. Hence, the z-score of M is +1.00. (Remember: concepts that apply to the standard deviation in distributions of scores apply to the standard error when we are assessing sample means, because the standard error is the standard deviation of the distribution of sample means (for samples of a given size)).)

If a sample of n = 9 scores is obtained from a population with mu = 70 and sigma = 18, then what is the z-score corresponding to a sample mean of M = 76? z = +1.00 z = +0.50 z = +3.00 z = +0.33

z = +1.00 (The z-scores of sample means must be assessed with respect to the distribution of sample means of the same size; for samples of n=9, the standard error here is sigma/sqrt(n) = 18/sqrt(9) = 6. The sample mean obtained is six points above the population mean, which is one standard error above the population mean. Hence, the z-score of M is +1.00. (Remember: concepts that apply to the standard deviation in distributions of scores apply to the standard error when we are assessing sample means, because the standard error is the standard deviation of the distribution of sample means (for samples of a given size)).)