I have 2 statistics problems that I've done and felt completely confident with them until I worked the problem a different way and got a different answer. I was wondering if you could tell me if I have the right answers or not. Thanks so much!
A certain air plane has 2 independent alternators to provide electrical power. The probability that one will fail is .02, what is the probability that.
(a) Probability both will fail = 0.0004
(b) Probability neither will fail (both will work fine) = 0.9604
(c) Probability one will fail = 0.0392
Problem 2:
A random sample of 10 miniature tootsie rolls was taken from a bag, each piece weighed on a very accurate scale. The results in grams were: 3.087 3.131 3.241 3.241 3.27 3.353 3.4 3.411 3.437 3.477
a)construct a 90% confidence interval for the true mean weight.
b)what sample size would be necessary to estimate the true weight with an error of .03 grams with 90% confidence.
c)discuss the factors which might cause variations in the weight of tootsie rolls during manufacturing.
Answers to a, b, and c:
A.
Sample mean 3.3048
Sample SD (s) 0.1320
Sample size (n) 10
Since sample size is smaller than 30, we will use a t-distribution.
Degree of freedom 9
Confidence level 90%
Critical tc 1.833
Margin of error 0.0765
Lower limit 3.2283
Upper limit 3.3813
B.
Margin of error 0.03
Confidence level 90%
Critical tc 1.833
SD 0.1320
Required sample size 65.04451373
Rounded up 6o 66.
C.
Factors include:machines that divide the raw material cannot divide perfectly equally. Although each was weighed on a very accurate scale there is still error of measurement. The temperature utilized during candy manufacture, sharpness of cutting tools utilized to form pieces.
Problem 2:
A random sample of 10 miniature tootsie rolls was taken from a bag, each piece weighed on a very accurate scale. The results in grams were: 3.087 3.131 3.241 3.241 3.27 3.353 3.4 3.411 3.437 3.477
a)construct a 90% confidence interval for the true mean weight.
b)what sample size would be necessary to estimate the true weight with an error of .03 grams with 90% confidence.
c)discuss the factors which might cause variations in the weight of tootsie rolls during manufacturing.
Answers to a, b, and c:
A.
Sample mean 3.3048
Sample SD (s) 0.1320
Sample size (n) 10
Since sample size is smaller than 30, we will use a t-distribution.
Degree of freedom 9
Confidence level 90%
Critical tc 1.833
Margin of error 0.0765
Lower limit 3.2283
Upper limit 3.3813
B.
Margin of error 0.03
Confidence level 90%
Critical tc 1.833
SD 0.1320
Required sample size 65.04451373
Rounded up 6o 66.
C.
Factors include:machines that divide the raw material cannot divide perfectly equally. Although each was weighed on a very accurate scale there is still error of measurement. The temperature utilized during candy manufacture, sharpness of cutting tools utilized to form pieces.
(a) To find the probability that both alternators will fail, you multiply the individual probabilities of each alternator failing together. Since the probability of one alternator failing is 0.02, the probability of both failing is 0.02 * 0.02 = 0.0004. So, you have the correct answer of 0.0004.
(b) To find the probability that neither alternator will fail (both will work fine), you subtract the probability of one alternator failing from 1. Since the probability of one alternator failing is 0.02, the probability of it not failing is 1 - 0.02 = 0.98. Since the two alternators are independent, the probability of both not failing is 0.98 * 0.98 = 0.9604. So, you have the correct answer of 0.9604.
(c) To find the probability that one alternator will fail, you need to consider two scenarios: the first one fails while the second one doesn't, or the second one fails while the first one doesn't. Since the probability of one alternator failing is 0.02, the probability of the first one failing and the second one not failing is 0.02 * 0.98 = 0.0196. Similarly, the probability of the second one failing and the first one not failing is also 0.02 * 0.98 = 0.0196. Adding these two probabilities together, you get 0.0196 + 0.0196 = 0.0392. So, you have the correct answer of 0.0392.
For problem 2:
(a) To construct a 90% confidence interval for the true mean weight, you can use the formula:
Margin of error = critical t-value * (sample standard deviation / square root of sample size)
The critical t-value depends on the desired confidence level and the degrees of freedom. Since the sample size is 10, the degrees of freedom are n - 1 = 10 - 1 = 9. For a 90% confidence level with 9 degrees of freedom, the critical t-value is 1.833. The sample mean is 3.3048, and the sample standard deviation is 0.1320. Plugging these values into the formula, you can calculate the margin of error as 0.0765. To construct the confidence interval, you subtract the margin of error from the sample mean to find the lower limit, and add it to the sample mean to find the upper limit. So, the confidence interval is (3.2283, 3.3813). Thus, you have the correct answer.
(b) To determine the sample size necessary to estimate the true weight with an error of 0.03 grams and a 90% confidence level, you can use the formula:
Sample size = (critical t-value)^2 * (sample standard deviation^2) / (desired error^2)
The critical t-value is 1.833 for a 90% confidence level with 9 degrees of freedom. The sample standard deviation is 0.1320, and the desired error is 0.03. Plugging these values into the formula, you can calculate the required sample size as 65.04451373. Since you cannot have a fraction of a sample, you round up to the nearest whole number, resulting in a required sample size of 66. So, you have the correct answer.
(c) The factors that might cause variations in the weight of tootsie rolls during manufacturing can include machines that divide the raw material not being able to divide perfectly equally, leading to variations in the individual weights. Additionally, even when weighing the pieces on a very accurate scale, there can still be measurement error. Other factors can include the temperature utilized during candy manufacture, which can affect the viscosity and consistency of the candy, leading to variations in weight. The sharpness of cutting tools utilized to form the pieces can also impact the weight consistency. So, you have the correct answer.