Question 1

A researcher records the following body mass index (BMI)scores for a group of participants: 28.4 ± 1.6 (M ± SD).Participants who score between 25 and 30 on this scale are categorized as overweight.What percentage of participants fell in this overweight category?

Accepted Answer

The answer of A researcher records the following body mass...

Question 2

State whether the first area is bigger,the second area is bigger,or the two areas are equal: the area to the left of z = 1.00,or the area to the left of z = -1.00.

Accepted Answer

The area to the left of z = 1.00 is bigger because the normal distribution is symmetric around the mean (z = 0), and areas to the left of positive z-scores are larger than those to the left of their negative counterparts.

Question 3

Which of the following is required to locate proportions of area under the normal curve?

Accepted Answer

To locate proportions of area under the normal curve, we need to first standardize the variable by computing a z-transformation. Then, we can use the unit normal table to find the probability of a z score corresponding to a certain area under the curve. Therefore, both steps are required, making choice C the best answer.

Question 4

A researcher records the following time spent sleeping per night (in hours)in a group of participants: 5.8 ± 1.8 (M ± SD).The researcher decides to categorize participants who sleep between 6 and 8 hr as moderate sleepers.What percentage of participants fell in this category?

Accepted Answer

To determine what percentage of participants are moderate sleepers (sleep between 6 and 8 hours), we need to calculate the number of participants falling within this range. 
6 hours (lower limit) minus the mean of 5.8 hours = 0.2/1.8 = 0.11 standard deviations above the mean. 
8 hours (upper limit) minus the mean of 5.8 hours = 2.2/1.8 = 1.22 standard deviations above the mean. 
Using a standard normal table, we can see that the percentage of participants falling within this range is approximately 34.13%. Therefore, the best choice is C, about 35%.

Question 5

A researcher records the following response times to a visual stimulus: 23 ± 3.1 (M ± SD)s.Assuming these data are normally distributed,what is the probability that participants responded in 20 s or less?

Accepted Answer

To find the probability of participants responding in 20 seconds or less, we need to standardize the value using the formula z = (x - μ) / σ. Here, x = 20, μ = 23, and σ = 3.1. Plugging in the values, we get z = (20 - 23) / 3.1 = -0.968. Looking up this value in the unit normal table gives us a probability of 0.1660, which is option A.

Question 6

Which of the following proportions is largest in a z distribution?

Accepted Answer

In a z distribution, which is a normal distribution standardized to have a mean of 0 and a standard deviation of 1, the proportion of area between z = +1 and z = -1 is approximately 68%. This is larger than the proportion of area greater than z = 1.0 (which is about 16%), less than z = 0 (which is 50%), or above the mean (also 50%).

Question 7

Which of the following is not a step to locate a proportion in a normal distribution?

Accepted Answer

Converting a proportion to a standard normal z score is not a step to locate a proportion in a normal distribution. The correct steps are to transform a raw score into a z score and then locate the corresponding proportion for the z score in the unit normal table.

Question 8

A counselor records the number of disagreements (per session)among couples during group counseling sessions.If the number of disagreements is distributed normally as 4.4 ± 0.4 (M ± SD)disagreements,then what proportion of couples disagree at least four times during each counseling session?

Accepted Answer

To find the proportion of couples who disagree at least four times, we calculate the Z-score for 4 disagreements using the formula Z = (X - M) / SD, where X is 4, M is 4.4, and SD is 0.4. This gives Z = (4 - 4.4) / 0.4 = -1. The proportion of couples disagreeing less than four times corresponds to the area to the left of Z = -1 in the standard normal distribution, which is 0.1587. Therefore, the proportion disagreeing at least four times is 1 - 0.1587 = 0.8413.

Question 9

Two normal distributions are observed.The first distribution has a mean of 12 and a standard deviation of 2.The second distribution has a mean of 0 and a standard deviation of 2.For which distribution will the z score for the mean be larger?

Accepted Answer

The z-score for the mean of a distribution is always 0 because the z-score formula is $Z = \frac{X - \mu}{\sigma}$, where $X$ is the value (in this case, the mean), $\mu$ is the mean, and $\sigma$ is the standard deviation. Substituting the mean for $X$ results in a z-score of 0 for any distribution, regardless of its mean or standard deviation.

Question 10

Which of the following is required to locate z scores with a given proportion?

Accepted Answer

To locate z scores with a given proportion, you need to find the z score associated with that probability and then compute a z transformation to find the raw score if needed.

Question 11

State whether the first area is bigger,the second area is bigger,or the two areas are equal: the area to the left of z = 0.80,or the area to the right of z = -0.80.

Accepted Answer

The area to the left of z=0.80 and the area to the right of z=-0.80 are symmetric to each other with respect to the mean of the standard normal distribution. Therefore, the two areas are equal.

Question 12

What is the z score for scores in the top and bottom 2.5%?

Accepted Answer

The z scores for the top and bottom 2.5% of a normal distribution are ±1.96. This is because a z score of ±1.96 corresponds to the 97.5th percentile and the 2.5th percentile, respectively, in a standard normal distribution.

Question 13

A researcher records the following number of birdcalls made during an experimental session with robins: 7.2 ± 0.6 (M ± SD).Assuming these data are normally distributed,what is the probability that robins made more than nine birdcalls during this experimental session?

Accepted Answer

To solve this problem, we need to use z-score formula:
z = (x - μ) / σ
where,
x = 9 (number of birdcalls we want to find)
μ = 7.2 (mean of birdcalls)
σ = 0.6 (standard deviation of birdcalls)
z = (9 - 7.2) / 0.6 = 3
Using the unit normal table, we can find the probability for z > 3, which is 0.0013. Therefore, the answer is C.

Question 14

A researcher records the following assessment data: 42 ± 6 (M ± SD).The researcher only wants to sample participants scoring in the top 20% of this assessment.What is the cutoff score for the top 20% in this example?

Accepted Answer

To find the cutoff score for the top 20%, we need to use the z-score formula. The z-score for the top 20% is 0.84. 
z-score = (X-M)/SD 
0.84 = (X-42)/6 
5.04 = X-42 
X = 47.04 
Therefore, the cutoff score for the top 20% is 47.04.

Question 15

A professor records the following final exam scores: 78 ± 7 (M ± SD).Students scoring in the top 14% get an A. What is the cutoff score for the top 14% in this example?

Accepted Answer

To find the cutoff score for the top 14%, we need to calculate the z-score corresponding to the 86th percentile (100% - 14%). Using a standard normal distribution table, we can find that the z-score corresponding to the 86th percentile is approximately 1.08. So, we can use the formula z = (x - M) / SD, and solve for x to find the cutoff score: x = z * SD + M = 1.08 * 7 + 78 = 85.56. Therefore, the cutoff score for the top 14% in this example is 85.56.

Question 16

A researcher measures the number of tasks completed by participants during a 5-min multitasking session.If the number of tasks completed is distributed normally as 6.3 ± 1.0 (M ± SD)tasks,then what is the probability that participants completed less than eight tasks?

Accepted Answer

To find the probability that participants completed less than eight tasks, we use the Z-score formula: Z = (X - M) / SD, where X is the value of interest (8 tasks), M is the mean (6.3 tasks), and SD is the standard deviation (1.0 task). Plugging in the values, we get Z = (8 - 6.3) / 1.0 = 1.7. Looking up the Z-score of 1.7 in the standard normal distribution table gives us a probability of approximately 0.9554, meaning there's a 95.54% chance that participants completed less than eight tasks.

Question 17

What is the z score for scores in the bottom 5%?

Accepted Answer

The z score for scores in the bottom 5% is -1.645 according to the z-score table.

Question 18

State whether the first area is bigger,the second area is bigger,or the two areas are equal: the area to the right of the mean,or the area between z = ±1.00.

Accepted Answer

The area between z = ±1.00 is approximately 68% of the total area under a standard normal curve. The area to the right of the mean (z > 0) is approximately 50% of the total area under the standard normal curve. Therefore, the area between z = ±1.00 is larger than the area to the right of the mean.

Question 19

A researcher records the following times (in seconds)that it took a group of mice to complete a maze: 23 ± 4.1 (M ± SD).The top 10% of mice finishing with the fastest times will be placed in a "maze bright" group.What is the cutoff time to be placed in this group?

Accepted Answer

To find the cutoff time for the top 10% of mice, we first need to calculate the z-score for 10% (or 0.10) using a standard normal distribution table or calculator. The z-score for 10% is approximately 1.28.

Then, we can use the formula: 
cutoff time = mean + (z-score * SD) 
cutoff time = 23 + (1.28 * 4.1) 
cutoff time ≈ 28.25 s

Therefore, the cutoff time to be placed in the "maze bright" group is 28.25 seconds.

Question 20

What is the z score for scores in the bottom 10%?

Accepted Answer

The z score for scores in the bottom 10% is -1.28. This is because z scores are standardized scores that indicate how many standard deviations an element is from the mean. A z score of -1.28 corresponds to the 10th percentile, indicating a value below the mean.

Quiz 6: Probability, normal Distributions, and Z Scores

A researcher records the following body mass index (BMI) scores for a group of participants: 28.4 ± 1.6 (M ± SD) .Participants who score between 25 and 30 on this scale are categorized as overweight.What percentage of participants fell in this overweight category?

State whether the first area is bigger,the second area is bigger,or the two areas are equal: the area to the left of z = 1.00,or the area to the left of z = -1.00.

Which of the following is required to locate proportions of area under the normal curve?

A researcher records the following time spent sleeping per night (in hours) in a group of participants: 5.8 ± 1.8 (M ± SD) .The researcher decides to categorize participants who sleep between 6 and 8 hr as moderate sleepers.What percentage of participants fell in this category?

A researcher records the following response times to a visual stimulus: 23 ± 3.1 (M ± SD) s.Assuming these data are normally distributed,what is the probability that participants responded in 20 s or less?

Which of the following proportions is largest in a z distribution?

Which of the following is not a step to locate a proportion in a normal distribution?

A counselor records the number of disagreements (per session) among couples during group counseling sessions.If the number of disagreements is distributed normally as 4.4 ± 0.4 (M ± SD) disagreements,then what proportion of couples disagree at least four times during each counseling session?

Two normal distributions are observed.The first distribution has a mean of 12 and a standard deviation of 2.The second distribution has a mean of 0 and a standard deviation of 2.For which distribution will the z score for the mean be larger?

Which of the following is required to locate z scores with a given proportion?

State whether the first area is bigger,the second area is bigger,or the two areas are equal: the area to the left of z = 0.80,or the area to the right of z = -0.80.

What is the z score for scores in the top and bottom 2.5%?

A researcher records the following number of birdcalls made during an experimental session with robins: 7.2 ± 0.6 (M ± SD) .Assuming these data are normally distributed,what is the probability that robins made more than nine birdcalls during this experimental session?

A researcher records the following assessment data: 42 ± 6 (M ± SD) .The researcher only wants to sample participants scoring in the top 20% of this assessment.What is the cutoff score for the top 20% in this example?

A professor records the following final exam scores: 78 ± 7 (M ± SD) .Students scoring in the top 14% get an A. What is the cutoff score for the top 14% in this example?

A researcher measures the number of tasks completed by participants during a 5-min multitasking session.If the number of tasks completed is distributed normally as 6.3 ± 1.0 (M ± SD) tasks,then what is the probability that participants completed less than eight tasks?

What is the z score for scores in the bottom 5%?

State whether the first area is bigger,the second area is bigger,or the two areas are equal: the area to the right of the mean,or the area between z = ±1.00.

A researcher records the following times (in seconds) that it took a group of mice to complete a maze: 23 ± 4.1 (M ± SD) .The top 10% of mice finishing with the fastest times will be placed in a "maze bright" group.What is the cutoff time to be placed in this group?

What is the z score for scores in the bottom 10%?