Sampling - Estimation of Means
Reference: Slides session 5-6.
1. Estimation of Means using Student t-distribution
Exercise 1. In reference to Exercise 4 in slide 17 (session 5-6), solve the following problem:
In a Pollution study of the air in a certain downtown area, an Environmental Protection Agency (EPA) technician obtained a mean of 3.5 micrograms of suspended benzene-soluble matter per cubic meter, with a standard deviation of 0.7 microgram for a sample of 16 measurements. Assuming that the population sampled is normal,
(a) Construct a 95% confidence interval for the mean of the population sampled. What's the maximum error of estimate of the mean?
(b) Construct a 97.5% confidence interval for the mean of the population sampled. What's the maximum error of estimate of the mean?
(c) Construct a 99% confidence interval for the mean of the population sampled. What's the maximum error of estimate of the mean?
Topic: Testing Hypothesis
Reference: Slides session 7. Exercises marked with a star (*) are optional but recommended.
1. TESTING HYPOTHESIS
Exercise 1
(a) What does testing a hypothesis mean? What is the general procedure?
(b) What is meant by type l and type errors?
(c) What is meant by the level of significance?
(d) What does the level of confidence mean?
Exercise 2
How can a producer of steel test that the breaking strength of fiber core (1 cm diameter)
wire rope produced is
(a) 1200 kg?
(b) Greater than 1200 kg?
(c) Less than 1200 kg?
2. TESTING HYPOTHESIS using the Normal Distribution
Exercise 3
AtlasMetal Inc., a producer of steel cables wants to test if the steel cables it produces have a breaking strength of 1200 kg. A breaking strength less than 1200 kg would not be adequate, and to produce steel cables with breaking strengths of more than 1200 kg would unnecessarily increase production costs. The producer takes a random sample of 64 pieces and finds that the average breaking strength is 1250 kg and the sample standard deviation is 240 kg. Should the producer accept the hypothesis that its steel cable has a breaking strength of 1200 kg at the 5% level of significance?
Hint 4. Set up the null and alternative hypothesis as follows, "Ho:µ = 1200 kg" and "H1: p. # 1200 kg"
Hint 5. Reminder: In case you accept the null hypothesis Ho and reject H1 at the 5% level of significance (or with a 95% level of confidence), this does not "prove" that [.t is indeed equal to 1200 kg. It only "proves" that there is no statistical evidence that p is not equal to 1200 kg at the 5% level of significance.
Exercise 4* In reference to Exercise 3, define the rejection and acceptance regions in terms of kg.
Hint 6. Use the formula: x¯ = μ ± z.S/√n 'here the value of z corresponds to a 95% confidence level.
Exercise 5 Cargo & Oceans Inc, is recruiting crew for a group of shipping companies. From past experience Cargo & Oceans nows that the weight of recruits is normally distributed with a mean i. of 75 kg and a standard deviation o-of 10 kg. The recruiting unit wants to test, at the 1% level of significance, if the average weight of this year's recruits is above 75 kg. To do this, it takes a random sample of 30 recruits and finds that the average weight for this sample is 80 kg.
(a) Formulate the null and alternative hypothesis
(b) Is it a right-tail test or two-tail test?
(c) Use the Table of Standard Normal Distribution to determine the tabular z-score corresponding to a 1% level of significance. Then determine the acceptance and rejection regions.
(d) Use available information to calculate the z-score
(e) Compare the calculated z-score and tabular z-score, and decide to accept or reject hypothesis at 1% level of significance.
Hint: This is a case you should use a one-tail test.
3. TESTING HYPOTHESIS using the Student's t Distribution
Exercise 6 Suppose that CereAlp Inc. wants to know if it can claim that the boxes of cereals it sells contain more than 500 g of cereals. From past experience the firm knows that the amount of cereals in the boxes is normally distributed. The firm takes a random sample of n = 25 and finds that = 520 g and s = 55 g. How can the firm conduct the test at the 5% level of significance?
Exercise 7* In reference to Exercise 6, define the rejection and acceptance regions in terms of the weight in grams.
Exercise 8 A consumer union receives many consumer complaints that the boxes of cereals sold by CereAlp Inc. contain less than the 560 g of cereals advertised. To check the consumers' complaints, the consumer union purchases nine boxes of the cereal and finds that I = 504 g and s = 84 g. How can the consumer union conduct the test at the 5% level of significance if it knows that the amount of cereal in the boxes is normally distributed?
Hint 7. You can use the left-tail test approach: null hypothesis Ho:µ = 560 g, with H1:µ < 560 g. Otherwise, with the null hypothesis Ho: p 560g, and H1: p < 560 gyou obtain the same result.