Question

A data set consisting of 320 men's basketball player heights is known to have a bell-shaped (normal) distribution with a mean of 75 inches and a standard deviation of 3 inches.

1)According to the Empirical Rule, approximately what percentage of the players' heights lie between the scores 69 and 81 inches?

• 99.7%
• 95%
• 68%
• 83

2)

The following table is the frequency distribution for the number of credit hours enrolled in by 18 students in a very large history class.

Credit Hours	Freq
0≤x<40≤x<4	2
4≤x<84≤x<8	9
8≤x<128≤x<12	2
12≤x<1612≤x<16	3
16≤x<2016≤x<20	2

Are these population or sample data? Select an answer Population Sample

The mean of these data is:

• 8.667
• 4.663
• 7.29
• 7.792

The variance of these data is:

• 36.378
• 24.471
• 23.111
• 27.722

The standard deviation of these data is:

• 5.265
• 6.031
• 4.807
• 4.947

Answer 1

1) By the empirical rule we know that 95% of the data for normal distribution lies within two standard distributions from the mean. We need the amount of data between 69 and 81. Note that 69 = 75 - 2*3 and 81 = 75 + 2*3. So 95% of te data will be between 69 and 81.

2) Since it is given that the data given is of 18 students in very large history class. So it is not population data but rather sample data as we do not have data for all te students in the class.

Here, we have intervals and not particular data points so for each interval we consider the mid-point to be the data point with the frequency corresponding to the interval.

x	m	f	m*f	f*(m-x̄)^2
0 <= x < 4	2	2	4	88.8888888898
4 <= x < 8	6	9	54	64.0000000016
8 <= x < 12	10	2	20	3.5555555554
12 <= x < 16	14	3	42	85.3333333323
16 <= x < 20	18	2	36	174.222222221
Total:		18	156	416.0000000001

x̄ = sample mean = (sum of m*f ) / 18 = 156 / 18 = 8.6666666667 so correct option 8.667

Sample variance = (sum of f*(m-x̄)^2) / (18 - 1) = 416 / 17 = 24.4705882353 so correct option 24.471

Sample standard dev = square root (variance) = 4.947

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