Question

estimate the average age at which multiple sclerosis patients were diagnosed with the condition for the first time in a given city. How big should the sample be? Define your procedures for this estimate (if necessary, set your own values of unknown parameters, based on statistical theory). In Table 1 you will find all ages of this patient population.

54 58 56 48 62 59 55 56 60 52
53 61 56 56 53 37 71 62 39 61
54 52 56 45 72 65 62 55 66 60
52 64 75 61 52 55 58 69 62 74
69 62 64 58 46 62 54 54 66 60
57 56 62 55 50 54 44 45 66 59
52 72 53 70 45 46 62 49 47 64
61 59 53 44 64 51 62 56 62 54
60 56 50 43 58 63 51 46 45 57
45 56 49 55 45 56 54 65 44 67
45 51 50 59 54 60 52 65 63 40
61 71 50 43 68 74 62 59 62 63
47 60 48 65 49 65 71 49 68 54
63 65 59 68 61 62 48 54 51 50
51 59 48 54 61 67 72 58 54 60
50 48 59 53 54 67 53 57 57 62
67 52 63 60 59 56 69 41 62 54
62 51 54 56 58 52 65 72 57 47
57 60 69 68 54 56 58 47 79 42
60 77 60 66 64 53 58 56 52 64
61 51 40 73 43 54 63 64 67 71
53 51 65 53 52 45 64 51 73 31
62 57 52 53 50 59 64 47 65 60
50 70 71 54 55 70 50 67 73 56
49 56 58 62 70 45 54 63 49 50
Table 1:  Ages in which multiple sclerosis patients are
diagnose the condition for the first time.

Answer 1

From the given data: Population size, N= 250

Maximum age =79

Minimum age =31

Estimate of Mean, $\bar{X}$=(maximum value+minimum value)/2 =(79+31)/2 =55 years

When the population size, N is known, the sample size, n required is as follows:

With 95% confidence level, we have the error level of E =1-95% =0.05. The minimum sample size, n is calculated as below:

n= N / (1 + N.E²) =250/(1+250(0.05)²) =154

However, usually sample size of 100 is preferred by many researchers when the population size is large. And the sample size of 30 or more is considered as a large sample.

Sample size(n) calculation using Margin of error: ​​​​​​​

Std.deviation, s =range/4 =(maximum - minimum)/4 =(79-31)/4 =12

Ler Margin of error, MoE =2 years

At 95% confidence level (two-tailed), Z =1.96

MoE =Z. s/$\sqrt{n}$$\implies$ n =[(Z.s)/MoE]² =[1.96(12)/2]² =139 (this method is used when population size is large).

By using this margin of error, we can construct confidence interval around the sample mean, $\bar{X}$​​​​​​.

Population mean, $\mu$ =$\bar{X}$$\pm$MoE

Now, let us calculate the actual mean of given 250 values:

The population Mean, $\mu$ =$\Sigma X/n$ =14328/250 =57.312 years