Question

Six data sets are presented, some of them are samples from a normal distribution, and some of them are samples from populations that are not normally distributed. Identify the samples that are not from normally distributed populations.

L1: Drug concentration six hours after administration

L2: Reading scores on standardized test for elementary children

L3: The number of minutes clerical workers took to complete a certain worksheet

L4: The level of impurities in aluminum cans (in percent)

L5: The number of defective items produced during an hour

L6: The weight of trout in ounces

L1	L2	L3	L4	L5	L6
4.7	72	4.5	2.1	21	9.9
3.2	77	5.8	1.3	16	11.3
5.1	65	3.7	2.8	10	11.4
4.7	85	4.9	1.5	10	9
3.0	68	4.3	1.0	11	10.1
3.4	83	4.7	8.2	9	8.2
4.4	73	5.8	1.9	13	8.9
3.5	79	3.2	9.5	12	9.9
4.5	72	3.0	3.2	11	10.5
5.8	81	5.1	1.3	29	8.6
3.7	79	3.6	4.4	10	7.8
4.9	91	4.3	3.8	14	10.8
5.4	69	3.6	2.7	27	8.4
3.6	67	5.4	8.0	10	9.6
4.3	82	4.7	1.9	11	9.9
3.0	66	3.0	4.9	11	8.4
5.1	77	3.4	4.5	12	9.0
4.3	66	4.3	1.5	19	9.1

For which of the sample data sets is it reasonable to treat as a sample from an approximately normal population?

		Drug concentration six hours after administration
		Reading scores on standardized test for elementary children
		The number of minutes clerical workers took to complete a certain worksheet
		The level of impurities in aluminum cans (in percent)
		The number of defective items produced in an hour
		The weight of trout in ounces

Answer 1

We'll use skewness to check for normal distribution here.

Skewness involves the symmetry of the distribution. Skewness that is normal involves a perfectly symmetric distribution. A positively skewed distribution has scores clustered to the left, with the tail extending to the right. A negatively skewed distribution has scores clustered to the right, with the tail extending to the left. Skewness is 0 (or near to 0 in practical cases) in a normal distribution, so the farther away from 0, the more non-normal the distribution.

Formula for skewness:

Calculation:

For skewness = 0,

Mean-Mode = 0 Mean = Median;

So, for checking normal distribution Mean should be equal to Median.

L1:

Mean = 4.25

Median = 4.35

Mean -  Median = 0.1 , which is nearing ,therefore, Drug concentration six hours after administration are normally distributed.

L2:

Mean = 75.11

Median = 75

Mean -Median= 0.11, which is nearing zero ,therefore, Reading scores on standardized test for elementary children are normally distributed.

L3:

Mean = 4.3

Median = 4.3

Mean Median, therefore the number of minutes clerical workers took to complete a certain worksheet are normally distributed.

L4:

Mean = 3.58

Median = 2.75

Mean Median, and Mean- Median = 0.83 which is not near zero, therefore The level of impurities in aluminum cans (in percent) are not normally distributed.

L5:

Mean = 14.22

Median = 11.5

Mean Median, and Mean- Median = 2.72 which is not near zero, therefore The number of defective items produced in an hour are not normally distributed.

L6:

Mean = 9.48

Median = 9.35

Mean - Median= 0.13, which is again nearing zero, therefore The weight of trout in ounces are normally distributed.

Hope this helps!