Question

1. The quality of the elementary school teachings is very important, and the effectiveness of teaching can be shown by the students’ performance on the standardized test. The state requires that the average score of the 5^th grade students from each school need to be at least 75. If any school’s average score is below 75, the school will be investigated further to ensure their quality of teaching. For School A, 20 5^th grade students are randomly selected to take the test, and their mean and standard deviation of the test score is 70 and 8.6 respectively. Analysis is needed to see if the average of all 5^th grade students in School A is below 75.

1. What is the population of the study? (Describe it specifically to the problem, don’t use general definition) .
2. What is the sample of the study? (Describe it specifically to the problem, don’t use general definition).
3. There are some numbers given from the study: 75, 70 and 8.6. Which numbers are sample statistic? Which numbers are population parameter?

2. A sample of the salaries of 10 employees of a small company are listed below, with unit of “thousand dollars”.

26

29

31

36

37

41

41

45

56

63

1. Find the mean and standard deviation of the sample data by hand.

xi	x	xi- x	(xi- x)2
26	40.5	-14.5	210.25
29	40.5	-11.5	132.25
31	40.5	-9.5	90.25
36	40.5	-4.5	20.25
37	40.5	-3.5	12.25
41	40.5	0.5	0.25
41	40.5	0.5	0.25
45	40.5	4.5	20.25
56	40.5	15.5	240.25
63	40.5	22.5	506.25
405			1232.5

2. Verify the calculations using Excel. What are the sample mean and sample standard deviation from Excel (use AVERAGE and STDEV functions)?
3. Find the 5-number summary of the data, and make a boxplot by hand.
4. Verify the 5-number summary found in (3) in Excel. What are the 5-number summary provided by Excel (use QUARTILE function)?

Answer 1

The population of the study - All 5th-grade students

The sample of the study is - 20 5th grade students from School A

The sample statistics are

i) 70 - Sample mean

ii) 8.6 - Sample standard deviation

The population parameter is

i) 75 - Population Mean

Mean of the sample

Sample standard deviation is given by

From the table, the sum of (Xi - X)^2 is = 1232.5

So, the sample standard deviation is

Calculation through excel gives the same result.

5-number summary of the data

Minimum Value - 26

First Quartile - 31

Median - (37+41)/2 = 39

Third Quartile - 45

Maximum Value - 63

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