Question

TOTAL MARKS: 25 QUESTION 3 (a) Periodically, the county council tests the drinking water of homeowners for contaminants such as lead. The lead levels in water specimens (ug/L) collected in 2017 for a sample of 12 residents of the county are shown below: 1.6, 3.3, 3.8, 0.7, 0.5, 1.3, 3.8, 4.2, 2.4, 2.9, 0.9, 3.5 (i) Calculate the sample mean and sample standard deviation for these measurements (ii) Construct the 99% confidence interval for the true mean μ using an appropriate distribution. (iii) Suppose that sixty rather than twelve different samples were measured and gave the same mean and standard deviation. Explain briefly why it would now be appropriate to use a different distribution and calculate the 99% confidence interval for the mean using this distribution 12 marks (b) Two batches of a certain chemical were delivered to a factory. For each batch, 15 determinations were made of the percentage of manganese in the chemical. The results were as follows: 2.6375, r22.4625, 0.2204, s20.2199 Is there a significant difference between the two sample means, testing at the 2% significance level? 13 marks)

Answer 1

I) from the given data, sample mean = 2.408

Sample SD =1.349

II) 99% confidence interval for u, is calculated using t score since population SD is unknown, so it will be

Sample mean +- t 0.005,n-1*sample sd/√n

= 2.408 +- 3.106*1.349/√12

=(1.145, 3.671)

III) Now that sample size is 60, we can use the z score since for n>30, the t distribution tends to the normal distribution. So the required confidence Interval is:

2.408 +- 2.576*1.349/√12

=(1.405, 3.411)

b) We are testing, H0: u1=u2 vs H1: u1 not equal u2

Pooled sample variance= 0.2204^2*14 + 0.2199^2*14/28 = 0.0485

So pooled standard deviation = √0.0485 = 0.2202

So standard error= 0.2202*√1/15 +1/15 = 0.0804

So test statistic : 2.6375-2.4625/0.0804= 2.176

P-value of this two sided t 28 test is:

2P(t28>2.176) = 0.03815

Since the p-value of this test> significance level of 0.02, we have insufficient evidence to reject H0 at the 2% significance level so we cannot conclude that there is a difference between the two means.