Question

Use the following collection of 30 test scores: 100, 65, 67, 61, 62, 70, 75, 73, 88, 77, 83, 79,95, 79, 8102, 86, 87, 87, 91, 87, 89, 92, 90, 99, 87, 72, 93, 79, 52

a. Construct a frequency distribution table using 6 classes: include classes, frequencies, class marks, class boundaries, relative frequencies, and relative percentages.

b. Find the mean to the nearest tenth and sample standard deviation to the nearest tenth.

c. Use the results from part b to determine the Z-score associated with a data value of 79.

Answer 1

Answer:-

Given That:-

Use the following collection of 30 test scores: 100, 65, 67, 61, 62, 70, 75, 73, 88, 77, 83, 79,95, 79, 8102, 86, 87, 87, 91, 87, 89, 92, 90, 99, 87, 72, 93, 79, 52

a. Construct a frequency distribution table using 6 classes: include classes, frequencies, class marks, class boundaries, relative frequencies, and relative percentages.

Class	Class marks	Frequencies	Relative Frequencies	Relative percentages
50-59	54.5	1	0.033	3.3
60-69	64.5	4	0.133	13.3
70-79	74.5	8	0.267	26.7
80-89	84.5	9	0.200	20
90-99	94.5	6	0.067	6.7
100-109	104.5	2

b. Find the mean to the nearest tenth and sample standard deviation to the nearest tenth.

Mean = 81.6

Stanadard deviation = 12.4

c. Use the results from part b to determine the Z-score associated with a data value of 79.

Z - score = (79 - 81.6)/12.4

= -0.21

(Mean and standard deviation are computed below)

The mean value of the data set is:

$\mu=\frac{2447}{30}\approx 81.5667$

Explanation:

The mean of a data set is the sum of the terms divided by the total number of terms. Using math notation we have:

Mean = Sum of terms/Number of terms

In this example:

Sum of terms = 100 + 65 + 67 + 61 + 62 + -----+ 52 = 2447

Number of terms = 30

Mean = Sum of terms/Number of terms

Mean = 2447/30

To find standard deviation we use the following formula

$\sigma=\sqrt{\frac{\Sigma (x_i-\bar{X})^2}{n-1}}$

we willl compute this formula in 4 steps.

Step 1:

Find the mean  $(\bar{X})$

In this example   = 81.5667. (use this calculator for step by step explanation on how to find mean)

X

Step 2:

Create the following table.

data	data-mean	(data-mean)2
100	18.4333	339.78654889
65	-16.5667	274.45554889
67	-14.5667	212.18874889
61	-20.5667	422.98914889
62	-19.5667	382.85574889
70	-11.5667	133.78854889
75	-6.5667	43.12154889
75	-6.5667	43.12154889
73	-8.5667	73.38834889
88	6.4333	41.38734889
77	-4.5667	20.85474889
83	1.4333	2.05434889
79	-2.5667	6.58794889
95	13.4333	180.45354889
79	-2.5667	6.58794889
80	-1.5667	2.45454889
102	20.4333	417.51974889
86	4.4333	19.65414889
87	5.4333	29.52074889
87	5.4333	29.52074889
91	9.4333	88.98714889
87	5.4333	29.52074889
89	7.4333	55.25394889
92	10.4333	108.85374889
90	8.4333	71.12054889
99	17.4333	303.91994889
87	5.4333	29.52074889
72	-9.5667	91.52174889
72	-9.5667	91.52174889
93	11.4333	130.72034889
79	-2.5667	6.58794889
52	-29.5667	847.18974889

Step 3:

Find the sum of numbers in the last column to get.

$\sum (x_i-\bar{X})^2=4429.3667$

step 4:

Calculate  $\sigma$ using the baove formula.

$\sigma=\sqrt{\frac{\Sigma (x_i-\bar{X})^2}{n-1}}$

$\sigma=\sqrt{\frac{4429.3667}{30-1}}$

$\sigma\approx 12.3587$

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