Question 1

The left tail of a distribution of a continuous variable is pulled extremely to the left. Which of the following would you expect to find given the shape of the distribution?

Accepted Answer

When the left tail of a distribution is pulled extremely to the left, it indicates that the distribution is skewed to the left, which corresponds to a negative skewness statistic.

Question 2

A distribution of a continuous variable is extremely peaked. Which of the following would you expect to find given the shape of the distribution?

Accepted Answer

A distribution that is extremely peaked (more peaked than a normal distribution) is described as having positive kurtosis. Kurtosis measures the "tailedness" of the distribution. Positive kurtosis indicates a sharp peak (leptokurtic), which is what is described in the question. Skewness, on the other hand, measures asymmetry, not peak sharpness.

Question 3

Which of the following is indicative of a distribution that is positively skewed?

Accepted Answer

A positively skewed distribution, also known as right-skewed, is characterized by a longer tail on the right side of the distribution, indicating that C) A right tail that is substantially pulled to the right is indicative of a positively skewed distribution.

Question 4

Which of the following would be found with a platykurtic distribution?

Accepted Answer

A platykurtic distribution is characterized by a lower, flatter peak compared to a normal distribution, which translates to a negative kurtosis statistic.

Question 5

The range of prices of new condominium sales ranges from $145,000 to $2,376,000. Most of the prices are bunched together in mid to low $200,000. What does this suggest in terms of the shape of the distribution?

Accepted Answer

The distribution is positively skewed because most prices are clustered at the lower end, with a long tail extending towards the higher prices.

Question 6

Weights (in pounds and ounces) of newborns ranges from 3 pounds 7 ounces to 8 pounds 14 ounces. Most of the weights of newborns bunch together between mid-seven to mid-eight pounds. What does this suggest in terms of the shape of the distribution?

Accepted Answer

The distribution is likely to be negatively skewed because most newborns' weights are bunched at the higher end of the range, indicating that the tail of the distribution extends more towards the lower end.

Question 7

Which of the following is indicative of a distribution that has a skew value of +18.90?

Accepted Answer

A skew value of +18.90 indicates a positive skew, meaning the tail of the distribution is longer towards the right side. This is consistent with option C, where it mentions a right tail that is substantially pulled to the right.

Question 8

Which of the following is indicative of a distribution that has a skew value of -14.76?

Accepted Answer

A skew value of -14.76 indicates a significant left skew, meaning the left tail of the distribution is substantially longer or more stretched out than the right tail. This corresponds to option B.

Question 9

Which of the following is indicative of a distribution that has a kurtosis value of +7.25?

Accepted Answer

A kurtosis value of +7.25 indicates a leptokurtic distribution, which means the distribution has heavier tails and a sharper peak compared to a normal distribution (mesokurtic). This value does not directly inform about the skewness of the distribution.

Question 10

Which of the following is indicative of a distribution that has a kurtosis value of -11.56?

Accepted Answer

A kurtosis value of -11.56 indicates a platykurtic distribution, which has flatter tails and a wider, more moderate peak compared to a normal distribution (mesokurtic). Leptokurtic distributions have positive kurtosis values and are more peaked with fatter tails. Skewness is unrelated to kurtosis and describes the asymmetry of the distribution, not its peakedness or tail thickness.

Question 11

The population mean of a distribution that is approximately normal is 500 and the population standard deviation is 100. What is the mean of the standardized distribution for this variable?

Accepted Answer

The mean of a standardized distribution (z-scores) is always 0, regardless of the original mean and standard deviation of the distribution.

Question 12

The population mean of a distribution that is approximately normal is 72 and the population standard deviation is 9. What is the mean of the standardized distribution for this variable?

Accepted Answer

The mean of a standardized distribution (or z-scores) is always 0, regardless of the original distribution's mean and standard deviation.

Question 13

A distribution has a mean of $25,000 and a standard deviation of $5,000. Which of the following is correct based on applying the Empirical Rule?

Accepted Answer

The Empirical Rule states that approximately 68% of the data falls within one standard deviation of the mean. Here, one standard deviation is $5,000, so 68% of the distribution is between $20,000 ($25,000 - $5,000) and $30,000 ($25,000 + $5,000).

Question 14

Apply the Empirical Rule to a distribution that has a mean of 275 and a standard deviation of 36. Which one of the following is correct?

Accepted Answer

The Empirical Rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. For a mean of 275 and a standard deviation of 36, two standard deviations from the mean are 275 - 2(36) = 203 and 275 + 2(36) = 347, which means about 95% of the distribution is between 203 and 347.

Question 15

Applying the Empirical Rule to a distribution with a mean of 80 and standard deviation of 5, which one of the following is correct?

Accepted Answer

The Empirical Rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. Given a mean of 80 and a standard deviation of 5, two standard deviations from the mean (80 ± 2*5) gives the range 70 to 90, which encompasses about 95% of the data.

Question 16

Which one of the following is a correct statement based on a distribution with a mean of 750 and standard deviation of 25?

Accepted Answer

Option C is correct because it uses the empirical rule, which states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. A score of 775 is one standard deviation above the mean (750 + 25), so about 84% of the scores would be less than 775 (50% below the mean plus 34% within one standard deviation above the mean).

Question 17

A distribution of 400 scores is approximately normally distributed. The mean of the distribution is 50 and the standard deviation is 10. What percentage of scores are less than a value of 30?

Accepted Answer

To find the percentage of scores less than a value of 30 in a normally distributed set with a mean of 50 and a standard deviation of 10, we calculate the Z-score for 30. The Z-score is (30 - 50) / 10 = -2. A Z-score of -2 corresponds to a percentile of about 2.5%, indicating that approximately 2.5% of the scores are less than 30.

Question 18

A distribution is approximately normal with a population mean of 80 and population standard deviation of 9. Approximately what percentage of scores fall between 71 and 89?

Accepted Answer

Using the empirical rule for normal distributions, approximately 68% (or about 2/3) of the data falls within one standard deviation (σ) of the mean (μ). Here, one standard deviation from the mean ranges from 71 (80 - 9) to 89 (80 + 9).

Question 19

What is the percentile rank of the score 610 in a unit normal distribution that has a population mean of 500 and standard deviation of 100?

Accepted Answer

The percentile rank of a score in a normal distribution can be found by converting the score to a z-score and then finding the corresponding percentile. The z-score is calculated as (score - mean) / standard deviation. For a score of 610, the z-score is (610 - 500) / 100 = 1.1. Looking up 1.1 on a standard normal distribution table or using a calculator, we find that it corresponds to a percentile rank of approximately 87.

Question 20

What is the 80th percentile of a distribution that is approximately normally distributed with a population mean of 75 and variance of 16?

Accepted Answer

The 80th percentile of a normally distributed population can be found using the Z-score associated with the 80th percentile, which is approximately 0.84. Using the formula for converting Z-scores to raw scores (X = μ + Zσ), where μ is the mean (75) and σ is the standard deviation (√variance = √16 = 4), we get X = 75 + (0.84)(4) = 75 + 3.36 = 78.36, which is approximately 78%.

Quiz 4: Normal Distribution and Standard Scores