And sn2 denote the sample mean and variance for the sample...

and $s_n^2$ denote the sample mean and variance for the sample $x_1, \dots, x_n$ and let $\bar{x}_{n+1}$ and $s_{n+1}^2$ denote these quantities when an additional observation $x_{n+1}$ is added to the sample. (a) (2 points) Show how $\bar{x}_{n+1}$ can be computed from $\bar{x}_n$ and $x_{n+1}$. (b) (6 points) Show that $ns_{n+1}^2 = (n-1)s_n^2 + \frac{n}{n+1}(x_{n+1} - \bar{x}_n)^2$. (c) (2 points) Ten batteries have a sample mean lifespan of 58 hours and a sample standard deviation of 2.3 hours. An 11th battery was tested to have a lifespan of 50 hours. What are the values of the sample mean and sample standard deviation for all 11 batteries?

Transcript

00:01 For this exercise, we are given a sample of size 10 that has a sample average of 40 and a half, and we are also told that the population's standard deviation is 1 .5.

00:12 For part a, we are asked if there's evidence from the sample to support the claim that the population mean exceeds 40, and we're asked to test this hypothesis at a significance level of alpha equals 0 .05.

00:32 So we're testing the claim that the mean is greater than 40.

00:37 So let's let the null hypothesis be that the mean is 40.

00:49 Now with alpha equals 0 .05, this is a one -sided, upper -tailed test, which means that our critical value is z sub 0 .05, and that is the 1 .645 figure.

01:13 And now we can make our test statistic.

01:15 We are given the population standard deviation, therefore our test statistic is z -0.

01:25 It's given by this formula.

01:27 So the mean here is the no hypothesized mean.

01:53 This comes out to 1 .265 approximately.

01:59 And we can see that our test statistic is less than our critical value.

02:09 Therefore we would fail to reject the no hypothesis.

02:22 For part b we are asked what the p value is for our test from part a.

02:27 The p value is the probability of getting a test statistic at least as extreme as the one we got.

02:36 Now in this situation this is an upper -tailed test, so we're looking for the probability of getting a test statistic greater than 1 .265.

02:43 It should be at least as extreme.

03:09 This comes out to a p -value of approximately .103.

03:24 Now for part c we are asked what the beta error is for the test in part a if the true mean is actually 42.

03:38 Now remember beta is the probability of failing to reject the no hypothesis when the null hypothesis is actually false.

03:49 This will be given by this formula.

04:05 This is an upper -tailed test, so this is positive, our critical value.

04:12 Delta is equal to the true population mean minus the no -hypothesized population mean.

04:20 And we have sample size and the population standard deviation from the question.

04:49 And this comes out to approximately 0 .0 .0 .0 .000.

04:52 0 .0032.

05:07 For part d, we are asked for what sample size would be required to ensure that beta does not exceed 0 .1 if the true mean is 44 hours.

05:38 Now, to find in in this situation, we use the formula as follows...

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