Section 3: The Lifetime of a certain type of Battery in...

Section 3: The Lifetime of a certain type of Battery in hours follows a gamma distribution with \(\alpha = 2\) and \(\beta = 1\). What is the probability that the Battery lasts between 4 and 6 hours? Also find the mean and variance of the lifetime of the Battery.

Transcript

00:01 There is given a normal distribution for this question, so the mean denoted by mu, that was given as 40 hours, and the standard deviation for the distribution which is 9.

00:09 So we can define the random variable x, which is normally distributed, the mean and the standard deviation.

00:15 So in the first part of the question, we have defined, we take randomly one battery of the production, which is x, so the probability of random variable x, which is greater than 30 hours.

00:27 To get this probability, i am going to use the graphing display calculator application, normalcdf, so the lower boundary is 30, there is no upper boundary, put a very big number, so the mean and the standard deviation.

00:37 Let's get this probability.

00:39 Press the second, and then variance, the normalcdf, so the lower boundary is 30, upper boundary is, this is 1, second is 99, and the mean is 40, and the standard deviation is 9, so the probability would be, which is 0 .86 and 67.

00:54 And for part b, so there is a sample size of 36, so the sample size given here is 36, so we need the sample mean.

01:03 The sample mean is the same with the population mean, which is 40, and also we need the sample standard deviation.

01:10 To get the sample standard deviation, we have to divide the population standard deviation by the root n, let's denote by sigma x bar, which is 9 over square root of 36, which is 9 over 6, this is 1 .5.

01:21 So i can define the random variable x bar, which is normally distributed, the mean and the standard deviation.

01:27 So we have to get the probability of this random variable x bar, which is between 37rs and 39rs.

01:35 Again, i'm going to use the normalcdf, so the lower boundary is 37, upper boundary is 39, and the mean is 40, and the standard deviation is 1 .5.

01:44 Press second, variance, the normalcdf, lower boundary is 37, upper boundary is 39, and the mean is 40, and the standard deviation is 1 .5.

01:54 So the probability would be 0 .22 and 97.

01:57 And for the next part of the question, this is another question not related with the previous question here...

Question

Section 3: The Lifetime of a certain type of Battery in hours follows a gamma distribution with \(\alpha = 2\) and \(\beta = 1\). What is the probability that the Battery lasts between 4 and 6 hours? Also find the mean and variance of the lifetime of the Battery.

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