MTH205 Introduction to Statistical Methods
Tutorial 2
Based on Chapter 2
1. Suppose that the weight of a bag of potato chips (in grams) is a normal random variable with an unknown mean µ and a known variance σ2 = 100. A random sample of 75 bags has mean x¯ = 50. Construct a 90% confidence interval for µ.
2. Calculate 90% confidence interval for the true mean density of the stock return from Apple company’s six monthly observations, namely,
4.50%, 5.60%, 4.81%, −2.67%, 5.76%, −4.15%.
What are the underlying assumptions?
3. In a study conducted, 20 oak seedlings were planted in the same type of soil and received the same amount of sunshine and water. Half of which, serving as a control, received no fertilizer while the other half received 368 ppm of nitrogen in the form. of sodium nitrate fertilizers. The stem weights, in grams, at the end of 140 days were recorded as follows.
No Fertilizer Fertilizer
0.32 0.26
0.53 0.43
0.28 0.47
0.37 0.49
0.47 0.52
0.43 0.75
0.36 0.79
0.42 0.86
0.38 0.62
0.43 0.46
Construct a 95% confidence interval for the difierence in the mean stem weight between seedlings that receive no fertilizer and those that receive 368 ppm of nitrogen. Assume the populations to be normally distributed with equal variances.
4. A manufacturer of car batteries claims that the batteries will last, on average, 3 years with a variance of 1 year. If 5 of these batteries have lifetimes of 1.9, 2.4, 3.0, 3.5 and 4.2 years, construct a 95% confidence interval for σ2 and decide if the manufacturer’s claim of σ2 = 1 is valid. Assume the population of battery lives to be normally distributed.
5. Two difierent brands of paints are being considered for use. Fifteen specimens of each type of paint were selected, and the drying times (in hours) were as follows.
Brand A Brand B
3.5 4.7
2.7 3.9
3.9 4.5
4.2 5.5
3.6 4.0
2.7 5.3
3.3 4.3
5.2 6.0
4.2 5.2
2.9 3.7
4.4 5.5
5.2 6.2
4.0 5.1
4.1 5.4
3.4 4.8
Construct a 95% confidence interval for σA
2 /σB
2 . Is it justifiable to assume equality of population variances?
6. A taxi company is trying to decide whether to purchase brand A or brand B tires for its fleet of taxis. To estimate the di↵erence in the two brands, an experiment is conducted using 12 of each brand. The tires are run until they they wear out. The respective sample means and standard deviations are given below.
Brand A Brand B
x¯A = 36300 km sA = 5000 km
x¯B = 38100 km sB = 6100 km
Construct a 95% confidence interval for µA − µB assuming the populations to be normally distributed. You may not assume that their variances are equal.
7. Grants are awarded to the agricultural departments of 9 universities to test the yield capabilities of two new varieties of wheat. Each variety was planted on a plot of equal area at each university and the yields, in kg per plot, were recorded as follows.
University
Variety #1 #2 #3 #4 #5 #6 #7 #8 #9
1 38 23 35 41 44 29 37 31 38
2 45 25 31 38 50 33 36 40 43
Find a 95% confidence interval for the main di↵erence between the yields of the two varieties, assuming the difierence of yields to be normally distributed. Explain why pairing is necessary in this problem.
8. In a random sample of 500 families owning television sets in the city of Hamilton, Canada, it is found that out of which 340 subscribe to HBO. Obtain a 95% confidence interval for p, the actual proportion of families with television sets in this city that subscribe to HBO. What is the minimum sample size required if we want to be 95% confident that our estimate of p is within 0.02 of the true value?
9. A certain change in a process for manufacturing component parts is being considered. Samples are taken under both the existing and the new process so as to determine if the new process results in an improvement. If 75 of 1500 items from the existing process are found to be defective and 80 of 2000 items from the new process are found to be defective, find a 90% confidence interval for the true difierence in the proportion of defectives between the existing and the new process. Based on this confidence interval constructed, comment if there is reason to believe that the new process produces a significant decrease in the proportion of defectives over the existing method.