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 (a) less than 22, (b) greater than 23, and (c) between 20 and 21.5? Assume sigma=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 less than 500? And assume sigma equals 75. And here we have 4 different answer choices labeled A through D. So, first and foremost, notice how our sample size or N. Is equal to 42, which is greater than 30. And because it's greater than 30. The sampling distribution of sample means approximates a normal distribution. So because of that, The mean of the sample means, or mu of X bar is just equal to mu, which is 530. So from here, we can find the standard deviation of the sample means. So, sigma of X bar. is equal to sigma divided by the square root of N. So that's 75 divided by the square root of 42. Which equals approximately 11.57. So what we want to find. Is the probability of X bar. Being less than 500. So first, we have to find the Z score that corresponds to 500. So Z is equal to. Xar Subtracted by view of X bar. And divided by sigma of X bar. So that is equal to 500, subtracted by 530. And divided by 11. 57. This gives you a Z score of approximately -2.592 for the score of 500. So, in actuality, the probability that we're looking for after converting the score of 500 into a Z score is P of Z being less than -2.592. So, when you reference the table to find the appropriate Z score, what you'll notice is that -2.592 is between -2.60 and -2.59. And so by interpolation, our probability is as follows. So P of Z being less than -2.592 is equal to -2.592. Subtracted by -2.60. Divided by -2.59. Subtracted by -2.60. Multiplied by 0.0048. Subtracted by 0.0047. And added to 0.0047. Notice how 0.0047 is. The probability corresponding to Z equals -2.60. And 0.0048 is the probability corresponding to Z equals -2.59. So after evaluating this expression, P of Z being less than. -2.592. Is approximately equal to 0.0048. And so now, that gave you your final answer, which corresponded to option A, the multiple choice. But before I end the video, I just wanted to say that there's another option available instead of interpolation. Because notice how the corresponding probabilities or -2.59. And -2.60, the Z scores are very, very close together. So because of that, since -2.592 is closer to -2.59, That probability could have instead been approximated as 0.048. But in any case, with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Central Limit Theorem

Standard Error of the Mean

Z-scores

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