Question

Find the mean, median, mode, and standard deviation (round standard deviation to the nearest whole number) for the following SAMPLE set of test scores.

            90 85 75 70 80 80

Using the above information and the Empirical Rule:

1. What percent of scores would you expect to fall between 66 and 87?                    ____________

2. 68% of the time you would expect scores between what two values?                     ____________

3. Determine the z-score for a test score of 64.                                                             ____________

4. What data value would have a z-score of + 2?                                                          ____________

5. Determine the z-score for a data value of 70 (round to nearest hundredth).           ____________

6. Why can we use the empirical rule with this data set? Explain your answer.

Let’s assume that another student scores a 95 on the next test. What would this do to the shape of the distribution? Explain your answer.

Answer 1

Dear student, please comment in the case of any doubt and I would love to clarify it.

Mean = 80, Median = 80. Mode = 80, S.D = 7.07 (Shown in excel)

a)

Anwer will be 81.5%.

b)

The empirical rule also referred to as the three-sigma rule or 68-95-99.7 rule, is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ). Broken down, the empirical rule shows that 68% falls within the first standard deviation (µ ± σ), 95% within the first two standard deviations (µ ± 2σ), and 99.7% within the first three standard deviations (µ ± 3σ).

Hence, for 68%, we need to calculate (µ ± σ).

(80 - 7.07) and (80 + 7.07), which will be 72.93 and 87.07.

c)

	=	standard score
	=	observed value = 64
	=	mean of the sample = 80
	=	the standard deviation of the sample = 7.07

z = (64 - 80) / 7.07 = -16 / 7.07 = -2.263

The z-score will be -2.263.

d)

Use z-score formula again

2 = (x - 80) / 7.07

x = 94.14

e)

z = (70 - 80) / 7.07 = -1.41

f)

Normal distributions come up time and time again in statistics. A normal distribution has some interesting properties: it has a bell shape, the mean and median are equal, and 68% of the data falls within 1 standard deviation.

Test scores follow the normal distribution.

The shape of the curve will be bell shaped if a student scores 95 marks.