Question 1

If the research sample contained 500 students, approximately how many had 21 or fewer friends?

Accepted Answer

The problem implies that a small percentage of the sample had 21 or fewer friends. Assuming a normal distribution, 33 out of 500 is a plausible estimate for a lower percentile.

Question 2

What does it mean when we say that a measurement instrument such as the SAT is a standardized test?

Accepted Answer

A

Question 3

A leptokurtic distribution of a variable:

Accepted Answer

A leptokurtic distribution has a steep peak, indicating that the data is more clustered around the mean than a normal distribution. It is not necessarily symmetrical and can have long tails.

Question 4

A z-score is:

Accepted Answer

A z-score represents how far a particular data point is from the mean in terms of standard deviations. It is calculated by subtracting the mean from the raw score and dividing by the standard deviation.

Question 5

In a research sample of high school students, the variable "number of MySpace friends" formed a normal distribution with a mean of 45 and a standard deviation of 16. Valentina had 53 friends. What was her approximate percentile rank?

Accepted Answer

We need to find the percentile rank of Valentina, who has 53 friends.

First, we need to standardize her score by finding the z-score:

z = (x - mean)/standard deviation
z = (53 - 45)/16
z = 0.50

Using a standard normal table, we can find that the area to the left of z = 0.50 is approximately 0.6915 or 69.15%.

Therefore, Valentina's approximate percentile rank is 69th.

Question 6

When statisticians refer to the distribution of a variable as "skewed," they mean that a frequency polygon for the variable:

Accepted Answer

A skewed distribution is one that is not symmetrical, may not be bell-shaped, and could contain outliers, all of which contribute to its lack of symmetry.

Question 7

Which of the following z-scores would fall at the lowest percentile?

Accepted Answer

A z-score of -1.35 has a lower value than the z-scores of 0.00, 1.35, and 2.00, meaning it corresponds to a lower percentile.

Question 8

Approximately what percentage of scores fall within one standard deviation of the mean?

Accepted Answer

According to the empirical rule, approximately 68% of scores fall within one standard deviation of the mean in a normal distribution.

Question 9

In the standard normal distribution:

Accepted Answer

A) The standard normal distribution is symmetrical.
B) The mean, median, and mode are all at the center of the distribution, which is at 0. 
C) The standard normal distribution approaches, but never touches the x-axis, as it is a continuous probability distribution.

Question 10

One hundred students took a statistics mid-term exam. The scores formed a normal distribution with a mean of 84 and a standard deviation of 6. Elice's grade was a 92. What was her approximate percentile rank?

Accepted Answer

To find Elice's percentile rank, we need to find the percentage of students who scored lower than her. We can use the Z-score formula to standardize Elice's score: Z = (X - μ) / σ = (92 - 84) / 6 = 1.33. We can then use a Z-score table or calculator to find the percentage of students below a Z-score of 1.33, which is approximately 91%. Therefore, Elice's approximate percentile rank is 91st.

Question 11

Approximately what percentage of scores of a normal distribution fall below -2 standard deviations?

Accepted Answer

Approximately 2% of scores in a normal distribution fall below -2 standard deviations, as the empirical rule (or 68-95-99.7 rule) indicates that about 95% of the data falls within two standard deviations of the mean, leaving 5% in the tails, with 2.5% in each tail.

Question 12

Lucille took the same exam, but received a 78 on it. What was her approximate percentile rank?

Accepted Answer

The answer of Lucille took the same exam, but received...

Quiz 4: Normal Distributions