Question

1. Frequency and Relative Frequency Distribution. 

What is the shape of the distribution of your sample data? 

Which class best estimates the center of the distribution? 

Do you have any modes? Are there any outliers? 

Make a guess as to the shape of the population distribution from the above histogram.

Use 2 decimal places for all values. 

Confidence interval Construction 

Validate the Assumptions/Conditions to construct a 95% Confidence interval. 

What is the symbol and value of the critical value for this interval? (You must interpolate.) Write the formula used and show work to calculate. 

What is you 95% CI? Fill in the table to the right above. 

Interpret your confidence interval in the context of this problem. 

What does 95% Confidence mean?

46
45
54
53
60
68
70
54
44
43
66
60
54
71
43
48
41
63
47
54
48
66
52
58
55
49
50
71
73
54
41
56
58
53
50
49
42
71
47
50
44
66
74
69
63
59
59
55
62
48

Answer 1

Frequency and relative frequency distribution -

Class	Frequency	Relative frequency
40-44	7	0.14
45-49	9	0.18
50-54	11	0.22
55-59	7	0.14
60-64	5	0.1
65-69	5	0.1
70-74	6	0.12
75-79	0	0

Frequencies for different cost groups

12 11 10 8- 7 6 6 5 5 4 2- 0 0+ 40 45 50 55 60 65 70 75 80 LO LO CO

Most of the values are clustered to the right side of the distribution.

Hence, the distribution is slightly right (or positive) skewed.

As, we can see from the histogram, class 50 - 55 best estimates the center of the sample distribution

Mode of the sample is 54 as it occurs most (5) times

The sample size is 50, we can compare the sample distribution to population distribution.

A rule of thumb is as soon as the sample size reaches 30, the sample distribution starts taking the shape of population distribution.

Base on this rule of thumb, the population distribution is also slightly right skewed

Outlier detection -

Mean:	55.5
SD:	9.5
# of observations:	50
Outlier detected?	No
Significance level:	0.05 (two-sided)
Critical value of Z:	1.96

(Critical value of z is looked up from student's z table)

Observation	Cost	Z	Significant Outlier?
1	46.	1.01
2	45.	1.11
3	54.	0.16
4	53.	0.27
5	60.	0.47
6	68.	1.32
7	70.	1.53
8	54.	0.16
9	44.	1.22
10	43.	1.32
11	66.	1.11
12	60.	0.47
13	54.	0.16
14	71.	1.64
15	43.	1.32
16	48.	0.79
17	41.	1.53
18	63.	0.79
19	47.	0.90
20	54.	0.16
21	48.	0.79
22	66.	1.11
23	52.	0.37
24	58.	0.26
25	55.	0.05
26	49.	0.69
27	50.	0.58
28	71.	1.64
29	73.	1.85
30	54.	0.16
31	41.	1.53
32	56.	0.05
33	58.	0.26
34	53.	0.27
35	50.	0.58
36	49.	0.69
37	42.	1.43
38	71.	1.64
39	47.	0.90
40	50.	0.58
41	44.	1.22
42	66.	1.11
43	74.	1.95	Furthest from the rest, but not a significant outlier (P > 0.05).
44	69.	1.42
45	63.	0.79
46	59.	0.37
47	59.	0.37
48	55.	0.05
49	62.	0.68
50	48.	0.79

Summary statistics -

Mean:	55.5
SD:	9.5
# of observations:	50
Standard error	1.35

Standard error = std dev / sqrt (sample size-1)

Point estimate	55.5
Margin of error	2.7
Lower bound	52.8
Upper bound	58.2

Where margin of error = t value * standard error

(t value for 49 df and 0.05 significance level for a two tailed hypothesis )

point estimate = mean

lower bound = point estimate - margin of error

upper bound = point estimate + margin of error

Hence, the 95% CI is

52.8 to 58.2

Interpretation of the 95% CI -

(95 sample means out of 100 are estimated to be within 52.8 to 58.2)

A 95% CI means -

95 estimates of the true population mean (sample mean) out of 100 estimates are estimated to lie between the calculated confidence interval