Question

A sample of size 30 is taken from a population and the height of each person is recorded.

4.8 4.8 4.0 5.2 5.5 4.7 5.6 5.5 4.9 5.0 5.2 4.6 5.6 4.9 5.1 5.2 5.2 5.5 4.3
5.2 5.2 5.2 4.7 5.3 4.8 4.4 4.4 5.2 4.6 5.1

1. Construct a frequency distribution table with 6 classes.

2. Plot a histogram and explain the shape of the distribution

3. Compute the mean and standard deviation of the data, using the frequency distribution.

4. What percentage of the observations falls within 1 standard deviations of the sample mean?

5. What percentage of the observations falls within 2 standard deviations of the sample mean?

6. Are the result of parts d and e coinciding with the empirical rule?

Answer 1

The frequency distribution table with 6 classes of given height of 30 person record is given below,

Frequency Table
Class	Frequency
4-4.2	1
4.3-4.5	3
4.6-4.8	7
4.9-5.1	5
5.2-5.4	9
5.5-5.7	5.

From the above frequency distribution table the histogram is given below,

From the above histogram we can say that the shape of histogram is approximately bell shaped. Please note that It is not perfectly bell shaped curve. But we can say that it is approximately bell shaped.

Now the summary is ,

Your Histogram
Mean	4.99
Standard Deviation (s)	0.40459
Skewness	-0.52471
Kurtosis	-0.17255
Lowest Score	4
Highest Score	5.6
Distribution Range	1.6
Total Number of Scores	30
Number of Distinct Scores	13
Lowest Class Value	4
Highest Class Value	5.7
Number of Classes	6
Class Range	0.3

  

By empirical rule 68% data will fall within one Standerd deviation. And 95% data will fall within Two Standerd deviation. And 99% of data will fall within three standard deviation.

In our results there is 70 % of data fall within one Standerd deviation and 96% of data fall within two Standerd deviation. so here the part d and e is approximately coinciding the Empirical rule.

And the shape also approximately bell shaped.

Thank you.