Question 1

Assume the average height for American women is 64 inches with a standard deviation of 2 inches.What percent of groups of size 25 would have mean heights of less than 62 inches? (Round z score to two decimal places.)

Accepted Answer

The standard error of the mean for a group of 25 is 2/sqrt(25) = 0.4. The z-score for a mean of 62 inches is (62-64)/0.4 = -5.00. A z-score of -5.00 corresponds to a probability of essentially 0%.

Question 2

Imagine that the mean height for all Division I women's basketball programs is 69 inches with a standard deviation of 3 inches.The 2010-2011 women's basketball team at the University of Connecticut, with 10 players listed on the roster, had an average height of 71.2 inches.Using the z statistic, what percent of means would fall below that for these UConn Huskies? (Round z score to two decimal places.)

Accepted Answer

First, we need to calculate the z score:

z = (x - μ) / (σ / √n)

Where x is the sample average height, μ is the population mean height, σ is the population standard deviation, and n is the sample size.

Plugging in the numbers:

z = (71.2 - 69) / (3 / √10) = 2.99

Looking at a standard normal distribution table, we can see that the percentage of means that fall below a z score of 2.99 is 0.998. Multiplying by 100 to get a percentage, we get 99.8%.

However, we need to take into account that we are looking for the percentage of means *below* the UConn Huskies' mean height of 71.2 inches. Since we're dealing with a normal distribution, we can flip the z score and look for the percentage of means above a z score of -2.99:

P(z > -2.99) = 1 - P(z ≤ -2.99) = 1 - 0.001 = 0.999

Multiplying by 100, we get 99.9%.

Finally, since we're looking for the percentage of means below the UConn Huskies' mean height, we need to subtract 99.9% from 100%:

100% - 99.9% = 0.1%

Rounding to two decimal places, we get 0.10%, which is closest to choice A (98.98%).

Question 3

To compare a group mean to a population, compute the:

Accepted Answer

The z statistic is used to compare a group mean to a population mean, assuming the population standard deviation is known. It measures how many standard deviations an element is from the mean.

Question 4

A research study that meets all the assumptions of the parametric test used to analyze the data:

Accepted Answer

A research study that meets all the assumptions of the parametric test used to analyze the data is likely to produce higher quality results than a research study that fails to meet some of the assumptions. This is because meeting the assumptions of a parametric test ensures that the data follows a normal distribution and has equal variances, which allows for more accurate and precise statistical testing. In contrast, a research study that fails to meet some of the assumptions may produce biased or unreliable results. However, it is important to note that even if a research study meets all the assumptions of a parametric test, it may still be beneficial to perform a nonparametric follow-up test to confirm the results.

Question 5

_____ requires that all members of a population have an equal chance of being selected for a study.

Accepted Answer

Random selection ensures that all individuals in the population have an equal chance of being chosen, which helps to reduce bias and increase the generalizability of the study results. Random assignment, on the other hand, refers to the process of randomly assigning participants to different groups or conditions in an experiment, which helps to control for extraneous variables. Normal distribution is a statistical concept related to the distribution of data points around a central mean, while the scale variable assumption pertains to the assumption that the measurement scales used in a study are appropriate for the research questions being asked.

Question 6

To convert a person's percentile rank into a raw score, one needs to know the mean and standard:

Accepted Answer

To convert a person's percentile rank to a raw score, one needs to know the score's position relative to the mean and standard deviation of the population. This allows us to determine which score corresponds to the given percentile rank.

Question 7

In one statistics course, students reported studying an average of 9.92 hours a week, with a standard deviation of 4.54.The professor tells David he scored at the 76th percentile.How many hours does he study per week? (Round z score to two decimal places.)

Accepted Answer

The z-score for the 76th percentile is 0.71. Using the formula z = (x - mu) / sigma, we can solve for x: 0.71 = (x - 9.92) / 4.54. Solving for x, we get x = 13.14 hours.

Question 8

Which statement is NOT an assumption of the z test?

Accepted Answer

The z test assumes the distribution of the population is approximately normal, not necessarily the sample. The Central Limit Theorem suggests that the sampling distribution of the sample mean will be approximately normal if the sample size is large enough, regardless of the shape of the population distribution.

Question 9

If a dependent variable is nominal, the assumption that the _____ should not be made.

Accepted Answer

Nominal data is categorical and cannot be measured using a scale measure. Therefore, assuming that the dependent variable is assessed using a scale measure is incorrect. The assumptions of normal distribution, random selection, and random assignment are not relevant to this particular issue.

Question 10

What are the consequences of failing to meet the assumptions of a parametric test when performing research?

Accepted Answer

Failing to meet the assumptions of a parametric test does not necessarily mean that we cannot make any inferences or that we are limited to qualitative descriptions of the data. If the test is robust to the violation, it does not invalidate the research, but it may make it more difficult to detect effects. However, if the violation is severe, we may need to consider using a nonparametric test instead.

Question 11

Assume the average height for American women is 64 inches with a standard deviation of 2 inches.What percent of groups of size 25 would have mean heights of more than 64.5 inches? (Round z score to two decimal places.)

Accepted Answer

The standard error (SE) is the standard deviation divided by the square root of the sample size: SE = 2/√25 = 0.4. The z-score for 64.5 inches is (64.5 - 64) / 0.4 = 1.25. Using the z-table, the probability of a z-score greater than 1.25 is approximately 10.56%.

Question 12

Because of the principle of _____, when sample sizes are at least 30, the distribution will most likely resemble a normal distribution.

Accepted Answer

The central limit theorem states that when sample sizes are at least 30, the distribution of the sample means will be approximately normally distributed, regardless of the underlying population distribution. This means that parametric statistics can be used to make inferences about the population parameters, as long as the sample size is large enough. Nonparametric statistics and robustness refer to different concepts and are not relevant to this question.

Question 13

Under what conditions is it permissible to proceed with a hypothesis test even though the assumption that participants are randomly selected is violated?

Accepted Answer

If participants are not randomly selected, the results may not be generalizable to the broader population. However, the hypothesis test can still be conducted with caution regarding the generalization of the results.

Question 14

The distribution of means has the same mean as the distribution of scores for the population, and the spread is:

Accepted Answer

The answer of The distribution of means has the same...

Question 15

Under what conditions is it permissible to proceed with a hypothesis test even though the assumption that the population distribution is approximately normal is violated?

Accepted Answer

When the sample size is greater than 30, it is generally safe to proceed with a hypothesis test even if the assumption of normality is violated. This is because of the central limit theorem, which states that as the sample size increases, the sampling distribution of the sample mean becomes approximately normal regardless of the shape of the population distribution.

Question 16

Inferential statistical analyses that are NOT based on a set of assumptions about the underlying population are:

Accepted Answer

Nonparametric tests are designed to be used when assumptions about the population are not met or when there is limited information about the population. These tests do not require assumptions about the underlying population and can provide a meaningful analysis of the data even when the sample is small, skewed, or contains outliers. Examples of nonparametric tests include the Wilcoxon rank sum test, Mann-Whitney U test, and Kruskal-Wallis test.

Question 17

Which statement is NOT an assumption of parametric hypothesis tests?

Accepted Answer

Parametric hypothesis tests do not assume that the sample size must be greater than 30. This is a general guideline for when the Central Limit Theorem kicks in, suggesting that the sampling distribution of the mean is approximately normal for large sample sizes, but it is not a strict assumption of all parametric tests.

Question 18

In one statistics course, students reported studying an average of 9.92 hours a week, with a standard deviation of 4.54.The professor tells Mackenzie she scored at the 46th percentile.How many hours does she study per week? (Round z score to two decimal places.)

Accepted Answer

To find the number of hours Mackenzie studies per week, given she is at the 46th percentile, we first need to find the z-score corresponding to the 46th percentile. The z-score for the 46th percentile is approximately -0.10 (since the 50th percentile, which has a z-score of 0, is the mean, and the 46th percentile is slightly below the mean). Using the z-score formula: $z = \frac{(X - \mu)}{\sigma}$, where $X$ is the value we're looking for, $\mu$ is the mean, and $\sigma$ is the standard deviation, we can rearrange the formula to solve for $X$: $X = z\sigma + \mu$. Substituting the given values: $X = -0.10(4.54) + 9.92$, we get $X \approx 9.47$ hours.

Question 19

Which statement regarding the denominator of the equation for the z score and that of the z statistic is true?

Accepted Answer

When computing a z score, we are comparing an individual score to the population mean using the population standard deviation. Therefore, we divide the difference between the individual score and the mean by the population standard deviation. 
However, when computing a z statistic, we are comparing a sample mean to the population mean. In this case, we divide the difference between the sample mean and the population mean by the standard error of the sampling distribution, which takes into account the variability of sample means. The denominator of the z statistic equation is thus different from that of the z score equation.

Question 20

Under what conditions is the assumption that the dependent variable is assessed on a scale measure NOT met?

Accepted Answer

Nominal or ordinal data do not have a clear numerical scale, so the assumption that the dependent variable is assessed on a scale measure is not met.

Quiz 7: Hypothesis Testing With Z Tests