1.The driveshaft lifespan is measured in operating hours before failure, and the distribution of lifespans is known to f
Posted: Mon May 23, 2022 10:33 am
1.The driveshaft lifespan is measured in operating hours before failure, and the distribution of lifespans is known to follow a Gamma distribution. A sample of 50 driveshafts is carefully tracked until failure and their lifespans are measured to have sample mean of 1274 hours, and a standard deviation of 108 hours.
(a)Assuming a normal distribution with the mean and standard deviation of the sample, estimate the proportion of driveshafts that would fail prior to reaching 1200 operating hours.
(b)Using the central limit theorem and corresponding normality assumption, calculate a 95% confi- dence interval for the true mean lifespan.
(c)Use the formulae αβ = x¯ and αβ2 = s2 to find values for the shape (α) and scale (β) parameters for a Gamma distribution with the same mean and standard deviation as the sample.
(d)Using this fitted Gamma distribution, estimate the proportion of driveshafts that would fail prior to reaching 1200 operating hours.
(e)Using the fitted Gamma distribution from (d) above, and the general facts that:
•If X1 ∼ ttamma(α1, β) and X2 ∼ ttamma(α2, β), with X1 and X2 independent of each other, then Y = X1 + X2 is such that Y ∼ ttamma(α1 + α2, β), and that
•If X ∼ ttamma(α, β) and Z = kX for a positive constant k > 0 then Z ∼ ttamma(α, kβ) and without making any other assumptions (without using the central limit theorem), estimate the distribution of the sample mean in this situation. Use that distribution to calculate an equal-tailed 95% confidence interval for the true mean lifespan.
(f)Compare your two estimates of the proportion of drive shafts that would fail prior to reaching 1200 operating hours ((a) and (d)), and your two confidence intervals for the mean lifespan ((b) and (e)). Comment on the similarity/ dissimilarity between making the normal assumption and using the gamma distribution.
(g)Which do you think is more useful: the estimate of the proportion of drive shafts that would fail prior to reaching 1200 operating hours, or the confidence intervals for
Assuming a normal distribution with the mean and standard deviation of the sample, estimate the proportion of driveshafts that would fail prior to reaching 1200 operating hours. (b) Using the central limit theorem and corresponding normality assumption, calculate a 95% confi- dence interval for the true mean lifespan. (c) As in Task 2, use the formulae aß = x and aß? =s to find values for the shape (a) and scale (B) parameters for a Gamma distribution with the same mean and standard deviation as the sample. (d) Using this fitted Gamma distribution, estimate the proportion of driveshafts that would fail prior to reaching 1200 operating hours. (e) Using the fitted Gamma distribution from (d) above, and the general facts that: • If X1 ttamma(Q1, B) and X2 ttamma az, B), with X1 and X2 independent of each other, then Y = X1 + X2 is such that Yttamma(a + a2, b), and that If X – ttammala, B) and Z=kX for a positive constant k > 0 then 2 ~ttamma(a, kB) and without making any other assumptions (without using the central limit theorem), estimate the distribution of the sample mean in this situation. Use that distribution to calculate an equal- tailed 95% confidence interval for the true mean lifespan. (f) Compare your two estimates of the proportion of drive shafts that would fail prior to reaching 1200 operating hours (a) and (d)), and your two confidence intervals for the mean lifespan (6) and (e)). Comment on the similarity/ dissimilarity between making the normal assumption and using the gamma distribution. (g) Which do you think is more useful: the estimate of the proportion of drive shafts that would fail prior to reaching 1200 operating hours, or the confidence intervals for the true mean lifespan? Explain your reasoning. .
(a)Assuming a normal distribution with the mean and standard deviation of the sample, estimate the proportion of driveshafts that would fail prior to reaching 1200 operating hours.
(b)Using the central limit theorem and corresponding normality assumption, calculate a 95% confi- dence interval for the true mean lifespan.
(c)Use the formulae αβ = x¯ and αβ2 = s2 to find values for the shape (α) and scale (β) parameters for a Gamma distribution with the same mean and standard deviation as the sample.
(d)Using this fitted Gamma distribution, estimate the proportion of driveshafts that would fail prior to reaching 1200 operating hours.
(e)Using the fitted Gamma distribution from (d) above, and the general facts that:
•If X1 ∼ ttamma(α1, β) and X2 ∼ ttamma(α2, β), with X1 and X2 independent of each other, then Y = X1 + X2 is such that Y ∼ ttamma(α1 + α2, β), and that
•If X ∼ ttamma(α, β) and Z = kX for a positive constant k > 0 then Z ∼ ttamma(α, kβ) and without making any other assumptions (without using the central limit theorem), estimate the distribution of the sample mean in this situation. Use that distribution to calculate an equal-tailed 95% confidence interval for the true mean lifespan.
(f)Compare your two estimates of the proportion of drive shafts that would fail prior to reaching 1200 operating hours ((a) and (d)), and your two confidence intervals for the mean lifespan ((b) and (e)). Comment on the similarity/ dissimilarity between making the normal assumption and using the gamma distribution.
(g)Which do you think is more useful: the estimate of the proportion of drive shafts that would fail prior to reaching 1200 operating hours, or the confidence intervals for
- 1 The Driveshaft Lifespan Is Measured In Operating Hours Before Failure And The Distribution Of Lifespans Is Known To F 1 (41.05 KiB) Viewed 14 times