Q1. The life in hours of a thermocouple used in a furnace is known to be approximately normally distributed, with standard deviation σ = 20 hours. A random sample of 15 thermo-couples resulted in the following data: 553, 552, 567, 579, 550, 541, 537, 553, 552, 546, 538, 553, 581, 539, 529.

(a) Is there evidence to support the claim that mean life exceeds 540 hours? Use a fixed-level test with α = 0.05.

(b) What is the P-value for this test?

(c) What is the β-value for this test if the true mean life is 560 hours?

(d) What sample size would be required to ensure that β does not exceed 0.10 if the true mean life is 560 hours?

(e) Construct a 95% one-sided lower CI on the mean life.

(f) Use the CI found in part (e) to test the hypothesis.

Q2. A post-mix beverage machine is adjusted to release a certain amount of syrup into a chamber where it is mixed with carbonated water. A random sample of 25 beverages was found to have a mean syrup content of x^- = 1.098 fluid ounces and a standard deviation of s = 0.016 fluid ounces.

(a) Do the data presented in this exercise support the claim that the mean amount of syrup dispensed is not 1.0 fluid ounce? Test this claim using α = 0.05.

(b) Do the data support the claim that the mean amount of syrup dispensed exceeds 1.0 fluid ounce? Test this claim using α = 0.05.

(c) Consider the hypothesis test in part (a). If the mean amount of syrup dispensed differs from μ = 1.0 by as much as 0.05, it is important to detect this with a high probability (at least 0.90, say). Using s as an estimate of σ, what can you say about the adequacy of the sample size n = 25 used by the experimenters?

(d) Find a 95% two-sided CI on the mean amount of syrup dispensed.

Q3. A normal population has known mean μ = 50 and variance σ² = 5. What is the approximate probability that the sample variance is greater than or equal to 7.44? Less than or equal to 2.56?

(a) For a random sample of n = 16.

(b) For a random sample of n = 30.

(c) For a random sample of n = 71.

(d) Compare your answers to parts (a)-(c) for the approximate probability that the sample variance is greater than or equal to 7.44. Explain why this tail probability is increasing or decreasing with increased sample size.

(e) Compare your answers to parts (a)-(c) for the approximate probability that the sample variance is less than or equal to 2.56. Explain why this tail probability is increasing or decreasing with increased sample size.

Q4. The warranty for batteries for mobile phones is set at 400 operating hours, with proper charging procedures. A study of 2000 batteries is carried out and three stop operating prior to 400 hours. Do these experimental results support the claim that less than 0.2% of the company's batteries will fail during the warranty period, with proper charging procedures? Use a hypothesis testing procedure with α = 0.01.

Q5. Identify an example in which a standard is specified or claim is made about a population. For example, "This type of car gets an average of 30 miles per gallon in urban driving." The standard or claim may be expressed as a mean (average), variance, standard deviation, or proportion. Collect an appropriate random sample of data and perform a hypothesis test to assess the standard or claim. Report on your results. Be sure to include in your report the claim expressed as a hypothesis test, a description of the data collected, the analysis performed, and the conclusion reached.

Q6. Two types of plastic are suitable for use by an electronics component manufacturer. The breaking strength of this plastic is important. It is known that σ₁ = σ₂ = 1.0 psi. From a random sample of size n₁ = 10 and n₂ = 12, we obtain x₁^- = 162.7 and x₂^- = 155.4. The company will not adopt plastic 1 unless its mean breaking strength exceeds that of plastic 2 by at least 10 psi. Based on the sample information, should it use plastic 1? Use the P-value approach in reaching a decision.