Download MATH 1040 Practice Problems: Confidence Intervals and Hypothesis Testing and more Exams Mathematics in PDF only on Docsity! MATH 1040 PRACTICE PROBLEMS FOR THE FINAL Remark 1. I will provide you with relevant tables on the actual exam, but you may use the ones in your text for the purposes of this midterm. 1. Problem 1 A simple random sample of size 50 is drawn from a population. The sample mean is 45 and the sample standard deviation is 16. Construct a 95% confidence interval about the mean. Solution 2. Since we do not know the population standard deviation we will use the t-distribution with 49 degrees of freedom. Since 49 does not show up on the table, we will use df = 50. We observe that α = 0.05 and hence tα 2 ≈ 2.009. Thus, our confidence interval is approximately defined by the bounds (1) 45± 2.009 · 16√ 50 . 2. Problem 2 Do we ever definitively accept the null hypothesis? Define the following terms: One-tailed test Two-tailed test Solution 3. In practice we do not ever definitively accept the null hy- pothesis. The only way that we could would be if we were to somehow get all the population data in which case it is somewhat unlikely that we would have needed to form the hypothesis in the first place. It should be pointed out that from a mathematical perspective the null hypothesis is unlikely to ever be true, at least for continuous data. With continuous data, one cannot measure the data exactly, and hence one is unlikely to measure parameters exactly. At this point, the differ- ence between statistically significant and practically significant comes into play. Let α be a (real-valued) parameter that we are testing. A two-tailed test would be a statistical test in which the hypotheses are written in the form H0 : α = α0 H1 : α 6= α0(2) 1 2 MATH 1040 PRACTICE PROBLEMS FOR THE FINAL where H0 is the null hypothesis and H1 is the alternative hypothesis. A one-tailed test would replace ‘ 6=’ in the alternative hypothesis with either ‘>’ or ‘<’. 3. Problem 3 Suppose that the null hypothesis is true but we reject the null hy- pothesis. What kind of error is that? Also, define the term ‘level of significance’. Solution 4. This is a type I error. The level of significance is the probability of making a type I error. 4. Problem 4 A simple random sample of size 50 is obtained from a population. The sample mean turns out to be 11.2 and the population standard deviation is known to be 2. Test the following claim: H0 : µ = 10 H1 : µ 6= 10.(3) The level of significance is 0.05. Solution 5. The test statistic is (4) z0 = 11.2− 10 2/ √ 50 ≈ 4.24 > 1.96 ≈ z0.025. We have sufficient statistical evidence (at the α = 0.05 level of signifi- cance) to reject the null hypothesis. 5. Problem 5 A simple random sample of size 50 is obtained from a population. The sample mean turns out to be 11.2 and the sample standard devi- ation is 1.5. Test the following claim: H0 : µ = 10 H1 : µ 6= 10.(5) The level of significance is 0.05.
