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Question 9 (1 point) Retake question Suppose that a sample of size 43 is drawn from a population distribution having mean 18 and standard deviation 11.6. with an un...

Question

Question 9 (1 point) Retake question Suppose that a sample of size 43 is drawn from a population distribution having mean 18 and standard deviation 11.6. with an unknown mean: Compute the Let X represent the sample following probability: P(x <19.008)0.2843 b. 0.6597 0.5346 d.0.7157 The probability cannot be determined from the given information Enter your answer as the lower-case letter associated with your choice.Previous PageNext PagePage 9 of 16

Question 9 (1 point) Retake question Suppose that a sample of size 43 is drawn from a population distribution having mean 18 and standard deviation 11.6. with an unknown mean: Compute the Let X represent the sample following probability: P(x <19.008) 0.2843 b. 0.6597 0.5346 d.0.7157 The probability cannot be determined from the given information Enter your answer as the lower-case letter associated with your choice. Previous Page Next Page Page 9 of 16



Answers

Answer the following questions for the sampling distribution of the sample mean shown. (a) What is the value of $\mu_{\bar{x}} ?$ (b) What is the value of $\sigma_{\bar{x}} ?$ (c) If the sample size is $n=16,$ what is likely true about the shape of the population? (d) If the sample size is $n=16,$ what is the standard deviation of the population from which the sample was drawn?

Now let's look at what information we can gather from uh diagram that they give us about a distribution here. They give us the diagram of the distribution of the sample means, notice the notation down here saying that it's experts distribution. So it's the sampling distribution of the sample means, oh first, what they're asking us for is what is the mean of the distribution of the sample means? Well from our diagram we see the 12 is right underneath the highest point of the curve, And so 12 is the population mean of our um sampling distribution of the sample means next, it wants us to give them the standard deviation of the distribution of the sample means. So here, from our sampling distribution, we see that this 12.05 is underneath where the curvature changes at a transition point, And also the 11.95 is at a transition point. And those values that are one transition point away from the mean is one standard deviation away from the mean. So we can find our standard deviation of our Sampling distribution by taking the 12.05 and subtracting the mean of 12.00 And we get 0.05, So are stigmas of x bar is 0.05. Now, if n is equal to nine, what is likely true about this shape of the distribution of the population? The population that we grab the samples out of? Well, notice from this diagram, they gave us that diagram is the diagram of a normal shape and it's from the sampling distribution of the X bars. Well, the sampling distribution of expire will have a normal shape if either it was, the samples were taken from a normal distribution to start with. If the sample size is small, Um if we had large sample size, then it population could have a shape that might not be normal but notice they're saying and is equal to nine. Remember nine is considered a small sample size, it's not considered a large sample size until your sample sizes 30 or more. So since an equal nine sample size is small and the distribution of X bar is approximately normal, that means that it's likely true that the distribution of the population had a shape that was normal. Yeah. And then lastly, if an equal nine find sigma, the population standard deviation. Well remember that sigma's of X. Bar can be found by taking sigma divided by the square root of N. We don't know sigma but we know sigma Cybex far from earlier in the problem that 0.5 And we know n. n is nine. So we have 0.5 is equal to sigma divided by the square root of nine. Well the squared of nine is three, so when I saw this equation, I'll just multiply both sides by three to solve for sigma. And we get the sigma is equal to, well, three times 0.50 point 15 Yeah.

Number 15 Christian E. Given the population distribution X is normal. If the population distribution X is normal, then the sampling distribution off off the sample mean X stage is also normal, so the answer will be yes. Question number be given it snowed to immune, equal 6.3 and X stage equal 4.5 and Zita equal three. Unequal 16. Does it? Value is a sample mean decreased boy. The population mean divide, divided by the standard deviation so that equal export minus new over zita over square root and equal 4.5 minus 6.3 over three over square root. 16. It almost equal minus point toe. Four. Question Number C. Given it snowed new equal 6.3 in each one in the tomb, you is less than 6.3 x dash equal 4.5. Zita equal three and unequal 16. Does it? Value is a sample mean decreased by the population mean divided by the standard deviation. The equal export minus mu minus New over. Zita over square root in equal 4.5, minus 6.3 over. See over square roots 16. It equal minus 2.4 the value is the probability off obtaining a value more extreme or equal to the standard standards. The test statics that did remind the probability using table five. So probability equal probability off that is less than minus 2.4 equals four point oh 82 with your number the given each node two new equal 6.3 and each one to mute is less than 6.3. Extension equal 4.5 Data equal three in equal 16. Goma Equal point is very Alfa equal points you one does it a value Is the sample mean decreased by the population being divided boy the standard deviation So the equal explore minus new over zita over a square square root in equal 4.5 minus 6.3 over three over squared 16 equal minus two points. Four. The B value is the probability off obtaining a value more extreme or equal to the standard Just static said did remind the probability. Using a table five. So probability equal probability off the is listen minus 2.4 equal 0.0 eat to If the if the value is smaller than the significance level Alfa, then the null hypothesis is rejected. Toe P is lesson 0.1 So a rejected reject h no

So in this question, were given a normally distributed population with men of 57.7 and a standard deviation, find the probability that a single randomly selected element X is less than 45. So this is a normally distributed population, so they can just convert that to our standard normal random variable, We have our population mean and standard definition. So that's probability oh z less than -1.052, which is .14 69 What? We were us to find the mean and standard deviation of the sample means that the mean of the sample mean is the population means Which is 57.7. The standard division is 12.1 over the square root of simple size, Which is 3.025. Find the probability that the mean of a sample of size 16 is less than 45, probability that the means of the sample is less than 45. So we take the sample means for for five minus 57 point 7/3 57.0.25 Which is equal to probably it easy, less than -4.19 that is zero.

Mhm. All right, that sounds are this question. Mhm. For a we see that the Mean value is just gonna be the same as the original mean value, which is 35. And for the standard evasion here, the sample size is nine. So the standard deviation of the sample means it's gonna be the sigma over square the band, which is 42 over square of nine. And the answer is 14. Probably we can do that because n is smaller than 30 and it's not. And if we don't make assumption that X is normal distribution, we cannot do that. All right. You can do it For semi size. That is 36. Gert, greater than 30. It's fine. Because regardless of the original distribution, we can always make assumption that the sample me is normally distributed. So the main value is 35 again, and and the standard evasion is Sigma Alpha Square van, which is 42 over square of 36, and the answer is seven. Okay, so, we got the answers.


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