The semester average grade for a statistics course is 76 with a standard deviati

Question

The semester average grade for a statistics course is 76 with a standard deviation of 5.5. Assume that stats grades have a bell-shaped distribution and use the empirical rule to answer the following questions (explain your responses with the help of a graph):

1. What is the probability of a student’s stat grade being greater than 87?

2. What percentage of students has stat grades between 70.5 and 81.5?

3. What percentage of students has stat grades between 70.5 and 65?

4. What is the probability of a student’s stat grade being greater than the mean?

Explanation / Answer

The empirical rule, also known as the three-sigma rule or the 68-95-99.7 rule, provides a quick estimate of the spread of data in a normal distribution given the mean and standard deviation. Specifically, the empirical rule states that for a normal distribution:

68% of the data will fall within one standard deviation of the mean.
95% of the data will fall within two standard deviations of the mean.
Almost all (99.7%) of the data will fall within three standard deviations of the mean.

mean = 76
sd = 5.5

1.
mean + 2*sd = 76 + 2*5.5 = 87
Hence 0.025 (i.e. 2.5%) is the probability of a student's stat grade being greater than 87.

2.
mean - 1*sd = 70.5
mean + 1*sd = 81.5

Hence, 68% of students has stat grades between 70.5 and 81.5

3.
mean - 1*sd = 70.5
mean - 2*sd = 76 - 11 = 65

Hence, (95/2 - 68/2) = 13.5% of students has stat grades between 70.5 and 65

4.
mean divides the bell shaped distibution in two equal parts.
Hence 0.5 is the probability of student's stat grade being greater than the mean.

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