Question 1

An English teacher has been teaching a sixth grade composition class for many years. He has the feeling that over the past several years, the writing ability of students has changed. A national test of proficiency in composition was administered 5 years ago. The resulting distribution of scores was normally shaped, had a mean of 85 and a standard deviation of 10.9. In order to test his feeling, he gives his present class of 43 students the same proficiency test. The resulting mean is 80 and the standard deviation is 8.7. Using the z test with a = 0.05 2 tail , what is your conclusion?

Accepted Answer

We need to use the z-test with a 0.05 significance level to compare the mean score of the present class with the mean score of the national test.

z-score = (80 - 85) / (10.9 / sqrt(43)) = -2.30

Looking at the z-table, we find that the area under the normal distribution curve beyond -1.96 and 1.96 is 0.05 (for a two-tail test).

Since the calculated z-score falls beyond -1.96, we reject the null hypothesis (H0: the present class is not deficient in composition writing) and conclude that the present class appears deficient in composition writing.

Question 2

If one draws all possible samples for various values of N from the same population of raw scores, as N increases _________.

Accepted Answer

The sampling distribution of the mean stabilizes as N increases, meaning that the mean of the sampling distribution of the mean stays the same. This is due to the Central Limit Theorem, which states that as sample size increases, the sampling distribution of the mean becomes more normal and its mean approaches the true population mean.

Question 3

If we take all possible samples from a normally distributed population, which of the following sample sizes will yield the smallest variation in the means of the samples?

Accepted Answer

Larger sample sizes tend to yield smaller variation in the means of the samples due to the Central Limit Theorem, which states that the standard error of the mean decreases as the sample size increases.

Question 4

Referring to question 29,how high must N be to be able to detect a µ real = 53 with a power of 0.8000?

Accepted Answer

The answer of Referring to question 29,how high must N...

Question 5

Referring to question 29, the probability of making a Type II error is _________.

Accepted Answer

The answer of Referring to question 29, the probability of...

Question 6

If one draws all possible samples for various values of N from the same population of raw scores, as N increases _________.

Accepted Answer

The standard error of the mean measures the variability of sample means when multiple samples are drawn from a population. As N increases, the variability of sample means decreases because larger sample sizes provide more representative estimates of the population mean. Therefore, the standard error of the mean decreases as N increases.

Question 7

An English teacher has been teaching a sixth grade composition class for many years. He has the feeling that over the past several years, the writing ability of students has changed. A national test of proficiency in composition was administered 5 years ago. The resulting distribution of scores was normally shaped, had a mean of 85 and a standard deviation of 10.9. In order to test his feeling, he gives his present class of 43 students the same proficiency test. The resulting mean is 80 and the standard deviation is 8.7. The obtained value of the appropriate statistic for testing H 0 is _________.

Accepted Answer

We are testing if there has been a significant change in the writing ability of students over the past 5 years, so our null hypothesis is:
H0: μ = 85 (there has been no change)

Our alternative hypothesis is:
Ha: μ < 85 (there has been a decrease in writing ability)

We are given the sample statistics:
n = 43
x̄ = 80
s = 8.7

To test our hypothesis, we can use a one-sample t-test, since the population standard deviation is unknown. The test statistic is given by:

t = (x̄ - μ) / (s / sqrt(n))

Plugging in the values, we get:
t = (80 - 85) / (8.7 / sqrt(43)) = -3.01

Looking up the t-distribution table with 42 degrees of freedom and a significance level of 0.05 (since it's a two-tailed test), we find the critical value to be -1.682. Since our test statistic (-3.01) is less than the critical value (-1.682), we reject the null hypothesis and conclude that there has been a significant decrease in writing ability over the past 5 years.

However, since the options in the question are all z-scores, we need to convert our t-score to a z-score using the formula:
z = (t - μt) / σt
where μt and σt are the mean and standard deviation of the t-distribution, respectively. Since we have 42 degrees of freedom, we can use the approximation that the t-distribution is approximately normal, so μt = 0 and σt ≈ 1.

Plugging in the values, we get:
z = (-3.01 - 0) / 1 ≈ -3.01

Therefore, the obtained value of the appropriate z-statistic for testing H0 is z obt = -3.01, which corresponds to option C.

Question 8

The critical value of the statistic _________.

Accepted Answer

The critical value of the statistic depends on alpha, which is the level of significance chosen by the researcher. This means that the critical value will be different for different levels of alpha.

Question 9

The critical value for the z distribution using a = 0.05 2 tail is _________.

Accepted Answer

For a two-tailed test with a significance level of 0.05, we need to find the z-score that encloses 95% of the area under the standard normal distribution. This critical value is ±1.96.

Question 10

The area under the critical region for rejection must _________.

Accepted Answer

The area under the critical region for rejection is defined to equal alpha, which is the significance level of the test. This area represents the probability of rejecting the null hypothesis when it is actually true.

Question 11

A set of sample scores from an experiment has an N = 28 and a mean = 52. H 1 is directional, predicting an effect of the independent variable which decreases the magnitude of the dependent variable. H 0 asserts the sample is a random sample from a population of scores where µ = 55 and s = 12. a = 0.05_{1 tail}. Using the z test to analyze the data, what is the correct conclusion regarding H 0 ?

Accepted Answer

The answer of A set of sample scores from an...

Question 12

An English teacher has been teaching a sixth grade composition class for many years. He has the feeling that over the past several years, the writing ability of students has changed. A national test of proficiency in composition was administered 5 years ago. The resulting distribution of scores was normally shaped, had a mean of 85 and a standard deviation of 10.9. In order to test his feeling, he gives his present class of 43 students the same proficiency test. The resulting mean is 80 and the standard deviation is 8.7. Using the z test with a = 0.05 2 tail , the appropriate critical value is _________.

Accepted Answer

For a two-tailed test with an alpha level of 0.05, the critical z values are ±1.96. This divides the tail of the normal distribution into two parts, each containing 2.5% of the total area (or probability).

Question 13

A set of sample scores from an experiment has an N = 28 and a mean = 52. H 1 is directional, predicting an effect of the independent variable which decreases the magnitude of the dependent variable. H 0 asserts the sample is a random sample from a population of scores where µ = 55 and s = 12. a = 0.05_{1 tail}. Using the z test to analyze the data, the power of the experiment to detect a real effect such that µ real = 53 is _________.

Accepted Answer

The answer of A set of sample scores from an...

Question 14

A set of sample scores from an experiment has an N = 28 and a mean = 52. H 1 is directional, predicting an effect of the independent variable which decreases the magnitude of the dependent variable. H 0 asserts the sample is a random sample from a population of scores where µ = 55 and s = 12. a = 0.05_{1 tail}. Using the z test to analyze the data, z obt = _________.

Accepted Answer

To use the z-test, we need to first calculate the standard error of the mean (SEM): SEM = s/√n = 12/√28 = 2.26. Then, we can calculate the z-score: z = (M - µ)/SEM = (52 - 55)/2.26 = -1.32. Since H1 is directional, we only consider one tail of the distribution (the left tail, in this case), and we find the critical z-value for α = 0.05 and a one-tailed test: z crit = -1.645. Since z obt = -1.32 > z crit = -1.645, we fail to reject H0.

Question 15

Power increases with _________.

Accepted Answer

Power increases with increasing N (sample size), increasing magnitude of the effect of the independent variable, increasing a level (alpha level) and the magnitude of the difference between µ real (true population mean) and µ null (hypothesized population mean). Therefore, all of these factors contribute to an increase in statistical power.

Question 16

A set of sample scores from an experiment has an N = 28 and a mean = 52. H 1 is directional, predicting an effect of the independent variable which decreases the magnitude of the dependent variable. H 0 asserts the sample is a random sample from a population of scores where µ = 55 and s = 12. a = 0.05_{1 tail}. Using the z test to analyze the data, z crit = _________.

Accepted Answer

Since H1 is directional, we are conducting a one-tailed test. We calculate the z-score using the formula: z = (M - µ) / (s / √N) = (52 - 55) / (12 / √28) = -1.23. To find the critical value, we look up the z-score associated with the 5% alpha level in a z-table: -1.645. Therefore, z crit = -1.645.

Question 17

Referring to question 29, if N is increased to 90, power = _________ to detect a real effect such that µ real = 53.

Accepted Answer

The answer of Referring to question 29, if N is...

Question 18

A raw score distribution that is negatively skewed will produce a sampling distribution of the mean for N = 4 which is _________.

Accepted Answer

The sampling distribution of the mean for a negatively skewed raw score distribution will also be negatively skewed, especially for a small sample size like N = 4. The Central Limit Theorem, which suggests that the sampling distribution of the mean will be approximately normal regardless of the shape of the population distribution, typically requires larger sample sizes to take effect.

Question 19

In cases where N > 1, the relationship between the raw score population standard deviation and the standard error is _________.

Accepted Answer

When N > 1, the standard error is always less than the standard deviation. The standard error is a measure of the uncertainty of the sample mean while the standard deviation measures the variability of data points in the entire population. As the sample size increases, the standard error decreases because the sample mean becomes a better estimate of the population mean.

Question 20

A raw score distribution which has a moderate negative skew will result in a sampling distribution of the mean for N = 42 that is _________.

Accepted Answer

According to the central limit theorem, regardless of the shape of the raw score distribution, as long as the sample size is sufficiently large (N = 42 is generally considered large enough) the sampling distribution of the mean will approach a normal distribution. Therefore, the answer is B) normally distributed.

Quiz 12: Sampling Distributions, Sampling Distribution of the Mean: the Normal Deviate Z Test

If one draws all possible samples for various values of N from the same population of raw scores, as N increases _________.

If we take all possible samples from a normally distributed population, which of the following sample sizes will yield the smallest variation in the means of the samples?

Referring to question 29,how high must N be to be able to detect a µ real = 53 with a power of 0.8000?

Referring to question 29, the probability of making a Type II error is _________.

If one draws all possible samples for various values of N from the same population of raw scores, as N increases _________.

The critical value of the statistic _________.

The critical value for the z distribution using a = 0.05 2 tail is _________.

The area under the critical region for rejection must _________.

Power increases with _________.

Referring to question 29, if N is increased to 90, power = _________ to detect a real effect such that µ real = 53.

A raw score distribution that is negatively skewed will produce a sampling distribution of the mean for N = 4 which is _________.

In cases where N > 1, the relationship between the raw score population standard deviation and the standard error is _________.

A raw score distribution which has a moderate negative skew will result in a sampling distribution of the mean for N = 42 that is _________.