Question

Score data from a statewide exam for 10th-graders follows a normal distribution, has a mean of 77, and has a standard deviation of 6.5. According to the Empirical Rule, 34% of test scores fall into what range? Select all that apply.

Answer 1

Given Information:  

Mean test score = μ = 77

Standard deviation of test score = σ = 6.5 seconds  

Answer:  

The lower limit represents those 34% test scores which are below the mean test score.

The range of test scores will be (70.5 to 77)

The upper limit represents those 34% test scores which are above the mean test score.

The range of test scores will be (77 to 83.5)  

Step-by-step explanation:  

Normal Distribution:

We are given a Normal Distribution, which is a continuous probability distribution and is symmetrical around the mean. The shape of this distribution is like a bell curve and most of the data is clustered around the mean. The area under this bell shaped curve represents the probability.

The Empirical Rule:

The empirical rule states that approximately 68% of all the data lie within 1 standard deviation from the mean, approximately 95% of all the data lie within 2 standard deviations from the mean and approximately 99.7% of all the data lie within 3 standard deviations from the mean.  

68% of all the data lie within 1 standard deviation from the mean, so that means 34% (half of 68%) of the test scores will be below the mean test score and remaining 34% of the test scores will be above the mean test score.

The confidence interval is given by

$CI = \mu \pm 1 \cdot \sigma \\\\CI = 77 \pm 1 \cdot (6.5) \\\\CI = 77 \pm 6.5 \\\\Lower \: limit = 77 - 6.5 = 70.5 \\\\Upper \: limit = 77 + 6.5 = 83.5$

The lower limit represents those 34% test scores which are below the mean test score.

So the range of test scores will be (70.5 to 77)

The upper limit represents those 34% test scores which are above the mean test score.

So the range of test scores will be (77 to 83.5)