In Exercises 55–60, find the indicated probabilities and inter...

Question

In Exercises 55–60, find the indicated probabilities and interpret the results.

The mean ACT composite score in a recent year is 20.7. A random sample of 36 ACT composite scores is selected. What is the probability that the mean score for the sample is (c) between 20 and 21.5? Assume σ=5.9.

Accepted Answer

All right. Hello, everyone. So this question says the mean SAT math score in a certain year is 530. A random sample of 42 SAT math scores is taken. What is the probability that the sample mean is between 527 and 532? Assume sigma is equal to 75. And here we have 4 different answer choices labeled A through D. So Recall it according to the central limit theorem. If random samples. Of side in Where N is greater than or equal to 30 are drawn from any population with a mean sigma and excuse me, with a mu and a standard deviation sigma, the sampling distribution of sample means approximates a normal distribution. So because and in this case is equal to 42. And that's greater than 30. Here, the sampling distribution approximates a normal distribution. Therefore mew. That is, the mu of sample means is equal to the population mean, or 530. So now we can use that information to calculate sigma. Sigma, if you recall. Or rather, sigma of X bar, which represents the sample means, is equal to sigma of the population divided by the square root of N. So that's 75 divided by the square root of. 42. And this equals approximately 11.57. So what are we looking for in this case? Well, what we want to find is the probability of X bar, which represents the sample means. For sample mean being between 5 and 27. And 532. So to find this range, we have to know the Z scores corresponding to both values. So let's start with 527. Recall that Z Is equal to x bar. Subtracted by mu of X bar. Divided by sigma of X bar. So That's 527. Subtracted by 530. Divided by 11.57. And this equals approximately negative 0.259. So now for 532. Z is equal to 532. Subtracted by 530. Divided by 11.57. And this equals approximately 0.173. So, therefore, another way of expressing what we're solving for is P of Z being between 0.259. And 0.173. So to find the range in between, or the probability falling between these two Z values, we have to subtract. And specifically, we have to subtract. P of Z less than 0.173. From he of. Excuse me, let me rephrase that. Because in actuality, I had the order backwards. So what we need to do is subtract P of. Z being less than 0.259. Bru, P of Z less than 0.173. So I just wanted to clarify that. And so, using the standard normal distribution table, we need to find both probabilities. But when referencing the table, you'll notice that both of these Z values fall between two values listed in the standard normal table. So, for both probabilities, you need to interpret it. Now, starting off, for example, with. -0.259. Notice how that value falls between. -0.25 and -0.26. So Their corresponding probabilities are as follows. For negative 0.25, that's 0.4013. And for negative 0.26, that's 0.3974. So now we can interpolate. So now P of. Z less than 0.259. Is equal to 0.259. Subtracted by 0.26. And divided by 0.25, subtracted by -0.26. This term is then multiplied by 0.4013. Subtracted by 0.3974. And then you add 0.3974. So when all is said and done, your probability is equal to approximately 0.3977. So now on to our next probability. So similarly 0.173 falls between two values on the standard normal table, specifically 0.17. And 0.18. So the probability Of less than 0.17. Is 0.5675. And for 0.18. It's 0.5714. So P of Z less than 0.173. Is equal to 0.173. Subtracted by 0.17. And divided by 0.18 subtracted by 0.17. This term is then multiplied by 0.5714. Subtracted by 0.5675. And then you add 0.5675 To give you a probability of approximately 0.5686. So now, to substitute the interpolated values. So he of. Z falling between 0.259. And 0.173. is equal to 0.5686, subtracted by 0.3977. And this equals approximately 0.1709. And there you have it. So your final answer is 0.1709. And if I scroll up here for just a moment. This corresponds to option C in the multiple choice, making that your correct answer. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

In Exercises 55–60, find the indicated probabilities and interpret the results.

The mean ACT composite score in a recent year is 20.7. A random sample of 36 ACT composite scores is selected. What is the probability that the mean score for the sample is (c) between 20 and 21.5? Assume σ=5.9.

Key Concepts

Central Limit Theorem

Standard Error of the Mean

Z-scores

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