ISE 261 HOMEWORK SEVEN Due Date: Thurs.5/03
1.Humerus bones from the same species of animal tend to have approximately the same length-to-width ratios. When fossils of humerus bones are discovered, archeologists can often determine the species of animal by examining the length-to-width ratios of the bones. It is known that a species of dinosaur has a mean ratio of 8.5. Suppose 41 fossils of humerus bones were unearthed at an archeological site in East Africa. East Africa is a location where this species of dinosaur is believed to have inhabited. (Assumethat the unearthed bones are all from the same unknown species). The length-to-width ratios of the bones were measured and the results are listed below.
Measured Results: x_bar = 9.257 and s = 1.203
Test the hypothesis that the unearthed bones are from this species of dinosaur. Since the significance level α, is also the probability of a Type I error, we will choose α to be very small because archeologists consider a Type I error to have very serious practical consequences; therefore,use α= 0.01.Provide a P-value with your analysis. Make a statement about this hypothesis test using the P-value.
Parameter:
Null Hypothesis:
Alternative:
Test Statistic:
Reject Region:
Calculation:
Decision:
P-value:
2.A random sample of 100 recorded deaths in the US during the past year showed an average life span of 71.8 years. Assuming a population standard deviation of 8.9 years, does this indicate that the mean life span today is greater than 70 years? Use a 0.05 level of significance. Provide a P-value with your analysis. Make a statement about this hypothesis test using yourP-value.
Parameter:
Null Hypothesis:
Alternative:
Test Statistic:
Reject Region:
Calculation:
Decision:
P-value:
3.Aircrew escape systems are powered by a solid propellant. The burning rate of this propellant is an important product characteristic. Specifications require that the mean burning rate must be 50 cm/s. The experimenter decides to specify a type I error probability of significance level of α = 0.05. He selects a random sample of n = 25 and obtains a sample average burning rate of x_bar = 51.3cm/s with sample standard deviation of s = 2 cm/s.Measurements indicate that burning rate follows a normal distribution. Test for conformance. What conclusion should he draw?
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4.Considering the rocket propellant in problem#3. What is the probability that the test will detect a different true mean (μ’) burning rate if μ’differs from 50 cm/s by as much as 1 cm/smore than 50 cm/sif the population standard deviation is actuallyσ = 1.25 cm/s. (Hint: d = |μ0 – μ’ | / σ )
Answer:
5.In evaluating a hypothesis test procedure in problem #1 with Humerus bones, archeologistshavea fundamental obligation to examine the probability of making a type II error, also denoted by β.Suppose that it is important for archeologists to reject the null hypothesis if the true mean ratio is μ’ =9.3.What is the probability that the archeologists will accept the null hypothesis when in fact the true mean ratio is μ’ =9.3.
(In most practical situations σ2 will be unknown. The sample variance can be substituted for s2 if the sample size n > 40.)
Answer:
6.An automatic filling machine is used to fill bottles with liquid detergent. A random sample of 20 bottles results in a sample variance of fill volume of s2 = 0.0153 square fluid ounces. If the variance of fill volume exceeds 0.01 square fluid ounces, an unacceptable proportion of bottles will be under or overfilled. Is there evidence in the sample data to suggest that the manufacturer has a problem with under and overfilled bottles? Use α = 0.05, and assume that fill volume has a normal distribution.
Parameter:
Null Hypothesis:
Alternative:
Test Statistic:
Reject Region:
Calculation:
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P-value: