Question

Where B = 2

Question 7 (10 marks) (a) A population of size 20 is sampled without replacement. The standard deviation of the population is 0.35. Find the minimum sample size that the standard error of the mean is no more than 0.15. (5 marks) (b) A sample of 30 masks was taken randomly from the production line. Assume the strengths of the marks follow a normal distribution. The strength of each specimen was determined, giving a sample standard deviation of 10 kN/m². With equal probabilities in the two tails, find the 95% confidence interval for the standard deviation of the strengths. (5 marks)

Answer 1

Solution:

a)

$\rightarrow$ Assuming population size is large

$\rightarrow$ Standard error of mean =  $\sigma /\sqrt{n}$ = 0.15

$0.35 /\sqrt{n}=0.15$

$n=5.44\approx \approx 6$

$\rightarrow$ Sample size = 6

b)

Given that :

$\rightarrow$ n=30

$\rightarrow$ Sample standard deviation s=10

$\rightarrow$ 95% confidence interval for standard deviation is

$CI=[\sqrt{\frac{(n-1)s^{2}}{x_{\alpha /2,n-1}^{2}}}<\sigma <\sqrt{\frac{(n-1)s^{2}}{x_{1-\alpha /2,n-1}^{2}}}]$

$=[\sqrt{\frac{29*100}{45.7223}<\sigma <\sqrt{\frac{29*100}{16.0471}}}]$

$=7.9641<\sigma <13.4431$

$\rightarrow$ 95% confidence interval for standard deviation is 7.96 to 13.44

$\Rightarrow 7.96<\sigma <13.44$

$\rightarrow$  Chi square critical values are obtained from chi square table with 29 degree of freedom.