1 do one of the following as appropriate a find the critical value z2 b find the critical value

Step 1: For the first part of the question, we need to find the critical value for a 90% confide

1 do one of the following as appropriate a find the critical...

1. Do one of the following, as appropriate: (a) Find the critical value zα/2, (b) find the critical value tα/2, (c) state that neither the normal nor the t distribution applies. 90%; n = 10; σ is unknown; population appears to be normally distributed. Group of answer choices A. zα/2 = 2.262 B. zα/2 = 1.383 C. tα/2 = 1.833 D. tα/2 = 1.812 2. Find the margin of error. 95% confidence interval; n = 91 ; = 53, s = 12.5 Group of answer choices A. 4.80 B. 2.34 C. 2.23 D. 2.60 3. Use the given degree of confidence and sample data to construct a confidence interval for the population mean μ. Assume that the population has a normal distribution. A laboratory tested twelve chicken eggs and found that the mean amount of cholesterol was 208 milligrams with s = 18.2 milligrams. Construct a 95 percent confidence interval for the true mean cholesterol content of all such eggs. Group of answer choices A. 196.5 < μ < 219.5 B. 196.3 < μ < 219.7 C. 196.4 < μ < 219.6 D. 198.6 < μ < 217.4 4. Find the critical value corresponding to a sample size of 6 and a confidence level of 95 percent. Group of answer choices A. 0.831 B. 12.833 C. 11.07 D. 1.145 5. Use the given degree of confidence and sample data to find a confidence interval for the population standard deviation σ. Assume that the population has a normal distribution. College students' annual earnings: 98% confidence; n = 9, = $3194, s = $821 Group of answer choices A. $646 < σ < $1071 B. $518 < σ < $1810 C. $499 < σ < $1607 D. $555 < σ < $1573

Transcript

00:01 Once again, welcome to a new problem.

00:05 This time we're dealing with confidence intervals.

00:09 We're dealing with confidence intervals.

00:16 So when you think about confidence intervals, we can have what you call confidence intervals for means, such that our confidence interval for means becomes x by plus minus t alpha over 2.

00:33 S of a radical n.

00:35 This portion is the margin error and within the context of the margin of error we have the sample size and we have the sample standard deviation so we have the sample size and the sample standard deviation and then we have the critical t value, you have the critical t value, and of course the sample mean.

01:16 And when it comes to confidence interval, we also have the confidence interval for population standard deviation.

01:32 We have the confidence interval for population standard deviation.

01:36 And it just so happens that the formula for the population standard deviation.

01:39 Confidence interval for population standard deviation is square root of n minus 1, s squared all of kai square alpha over 2, and radical n minus 1, s squared, kai square 1 minus alpha over 2.

02:05 The kye square values, these are the critical kai square value for the lower end and upper limit.

02:26 And then of course, s squared is the sample variance.

02:33 So s squared gives us the sample variance.

02:37 We're looking at a new problem and in this particular problem that we're looking at, the first part in number one, we are given an alpha of confidence interval of 90%.

02:56 And we want to determine the z alpha over 2 or the z critical value for 90 % confidence interval.

03:04 So in that sense, if you have a 90 % confidence interval, it means that we have negative x -a -4 -2 and positive z -alph over 2.

03:18 The alpha -over -2 is the same as these two portions that you're seeing, so 0 .025 and 0 .025.

03:28 And so the z alpha over 2 based on the z distribution table, so you have to go back to the z distribution table.

03:40 And our z alpha over 2 at 90 % will be the same as, so you're looking at 0 .025.

03:55 That's the number that you're looking at.

03:59 So we're going to get a 90 % confidence interval, a 90 % confidence interval.

04:13 So this is z -f over 2, and so looking at the 90 % confidence, we have 2.

04:27 So this is for the z score.

04:36 2.

04:43 So at 90%, we have noise 10%, so 0 .10 % divided by 2 .0 .05.

04:55 So this value right here, this value that we're looking at, this is 0 .05.

05:00 So we just want to make sure that it's the appropriate number.

05:04 This is 0 .05.

05:06 And of course, this 1 2 is 0 .05.

05:09 So when we're looking at z .05, yes, that's the one we're looking at, we're going to get 2 .6, 2 .05.

05:27 So 0 .05, not 2 .05, but 0 .05 is going to give us 1 .645 .5.

05:40 4 -5 so that's what we're looking at so this is for z a 4 over 2 at 90 % at 90 % so we have 10 % 10 divided by 100 divide by 2 .05 is the probability and so we get find the critical value for z a alpha over 2 for z r4 over 2 that's going to give us 1.

06:27 So we have 1 minus 0 .05 that's 0 .95 95 1 .645.

06:49 1 .645.

06:56 So that's the z score.

06:58 We're looking at the z score if you're looking at the z distribution table, once again this is 90 % confidence i'm assuming this is 90 % confidence and it's a normal distribution so the assumption is a normal distribution and then of course we also want to get the second part this is for t alpha over 2 that's the critical value so we have t alpha over 2 which is point 05 and the degrees of freedom we have a sample size of 10 so n minus 1 is 10 minus 1 which is 9 so combining these two the degrees of freedom and the critical value because we're looking at the t critical value for two -tailed test it's 10 % so at 9 we're going to have 1 .833.

08:19 That's at 9, it's going to be 1 .833.

08:25 So t rfa over 2 becomes 1 .833.

08:34 That's if we're using the t distribution.

08:43 So if we're using the t distribution, that's what happens.

08:47 And then we're looking at the second.

08:49 Part so we've done the distribution the t critical for the second part state the normal if it's normal or a t distribution applies so we're gonna say since since the sample size is less than equal to is less than 30 is less than 30 is less than 30 then the t distribution, the t distribution applies.

09:36 So because the sample size is less than 30, we're going to apply the t distribution in the particular problem.

09:45 And then in the second part of the problem, we want to get the 95 % confidence for these values...

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