1. A set of test scores is normally distributed with a mean of 76 points and a standard deviation of 8 points. Round your answers to the nearest percentile.
a. A student with a score of 76 points would be in the____th percentile.
b. A student with a score of 68 points would be in the ____th percentile.
c. A student with a score of 95 points would be in the_____th percentile.
2. Which of the following statements are TRUE about the normal distribution? Check all that apply.
a. The mean corresponds to the z-score of 1.
b. A z-score is the number of standard deviations a specific data value is from the mean of the distribution.
c. The area to the left of a z-score plus the area to the right of that same z-score will always equal 1.
d. The Empirical Rule only applies when a value is exactly 1, 2, or 3 standard deviations away from the mean.
e. A data value with z-score = -1.5 is located 1.5 standard deviations below the mean.
David Lane normal distribution
1. To find the percentiles, we can use the z-score formula:
z = (x - μ) / σ
where x is the value, μ is the mean, and σ is the standard deviation.
a. For a score of 76 points,
z = (76 - 76) / 8 = 0 / 8 = 0
To find which percentile this corresponds to, we can use a standard normal distribution table or a calculator. Since z = 0, the value falls at the 50th percentile.
b. For a score of 68 points,
z = (68 - 76) / 8 = -8 / 8 = -1
Using the same method, we find that the value falls at approximately the 16th percentile.
c. For a score of 95 points,
z = (95 - 76) / 8 = 19 / 8 ≈ 2.375
Using the same method, we find that the value falls at approximately the 99th percentile.
Therefore:
a. A student with a score of 76 points would be in the 50th percentile.
b. A student with a score of 68 points would be in the 16th percentile.
c. A student with a score of 95 points would be in the 99th percentile.
2. Let's go through each statement to determine which ones are true:
a. The mean corresponds to the z-score of 1.
This statement is not true. The mean corresponds to a z-score of 0. In a standard normal distribution, a z-score of 1 corresponds to being one standard deviation above the mean.
b. A z-score is the number of standard deviations a specific data value is from the mean of the distribution.
This statement is true. A z-score measures the number of standard deviations a specific data value is from the mean. It tells us how far away a particular value is from the mean in terms of standard deviations.
c. The area to the left of a z-score plus the area to the right of that same z-score will always equal 1.
This statement is true. In a standard normal distribution, the total area under the curve is equal to 1. This means that the probability of an event occurring is always 1 or 100%.
d. The Empirical Rule only applies when a value is exactly 1, 2, or 3 standard deviations away from the mean.
This statement is not true. The Empirical Rule, also known as the 68-95-99.7 rule, applies to values within 1, 2, or 3 standard deviations of the mean. It gives us a rough estimate of the proportion of data within these ranges.
e. A data value with z-score = -1.5 is located 1.5 standard deviations below the mean.
This statement is true. A negative z-score indicates that the data value is below the mean. In this case, a z-score of -1.5 means the data value is located 1.5 standard deviations below the mean.
Therefore, the true statements are:
b. A z-score is the number of standard deviations a specific data value is from the mean of the distribution.
c. The area to the left of a z-score plus the area to the right of that same z-score will always equal 1.
e. A data value with z-score = -1.5 is located 1.5 standard deviations below the mean.
1. To find the percentiles, we can use the z-score formula:
a. To find the percentile for a score of 76 points:
Z = (X - μ) / σ
Z = (76 - 76) / 8
Z = 0
Looking up the z-score of 0 in the z-score to percentile table, we find that it corresponds to the 50th percentile. Therefore, a student with a score of 76 points would be in the 50th percentile.
b. To find the percentile for a score of 68 points:
Z = (X - μ) / σ
Z = (68 - 76) / 8
Z = -1
Looking up the z-score of -1 in the z-score to percentile table, we find that it corresponds to the 16th percentile. Therefore, a student with a score of 68 points would be in the 16th percentile.
c. To find the percentile for a score of 95 points:
Z = (X - μ) / σ
Z = (95 - 76) / 8
Z = 2.375
Looking up the z-score of 2.375 in the z-score to percentile table, we find that it corresponds to the 99th percentile. Therefore, a student with a score of 95 points would be in the 99th percentile.
2. The TRUE statements about the normal distribution are:
b. A z-score is the number of standard deviations a specific data value is from the mean of the distribution.
c. The area to the left of a z-score plus the area to the right of that same z-score will always equal 1.
e. A data value with z-score = -1.5 is located 1.5 standard deviations below the mean.