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Numerade Educator

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Problem 5 Medium Difficulty

NAEP scores Young people have a better chance of full-time employment and good wages if they are
good with numbers. How strong are the quantitative skills of young Americans of working age? One source of data is the National Assessment of Educational Progress (NAEP) Young Adult Literacy Assessment Survey, which is based on a nationwide probability sample of households. The NAEP survey includes a short test of quantitative skills, covering mainly basic arithmetic and the ability to apply it to realistic problems. Scores on the test range from 0 to 500. For example, a person who scores 233 can add the amounts of two checks appearing on a bank deposit slip; someone scoring 325 can determine the price of a meal from a menu; a person scoring 375 can transform a price in cents per ounce into dollars per pound. $^{4}$ Suppose that you give the NAEP test to an SRS of
840 people from a large population in which the scores have mean 280 and standard deviation S 60. The mean $\overline{x}$ of the 840 scores will vary if you take repeated samples.
(a) Describe the shape, center, and spread of the sampling distribution of $\overline{x} .$
(b) Sketch the sampling distribution of $\overline{x}$ . Mark its mean and the values one, two, and three standard deviations on either side of the mean.
(c) According to the $68-95-99.7$ rule, about 95$\%$ of all values of $\overline{x}$ lie within a distance $m$ of the mean of the sampling distribution. What is $m ?$ Shade the region on the axis of your sketch that is within $m$ of the mean.
(d) Whenever $\overline{x}$ falls in the region you shaded, the population mean $\mu$ lies in the confidence interval $\overline{x} \pm m .$ For what percent of all possible samples does the interval capture $\mu$ ?

Answer

a. Approximately normal with mean 280 and standard deviation 2.0702
b. see drawing
c. $m=4.140$
d. 95$\%$

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Video Transcript

in question five. We have information about a certain type of test that young people can take in their scores. Scores range from 0 to 500 and we have a simple random sample of 840 people from a large population. We have a mean score of 280 and a standard deviation of 60 in part A. We were asked to describe the shape, center and spread of the sampling distribution in this case, Well, because we have a large population that indicates that our shape is approximately normal. Best central limit The room that comes into play here, our center. Because we have a sampling distribution, the center of our sampling distribution will ultimately match the population. Mean so in this case, our center of our sampling distribution. Thanks, Bar 280. And then the spread in this case s are standard deviation is 60. But in order to calculate our spread of a sampling distribution of explore, we have to take our population, senator vision and divide by route in, which is gonna be 60 about about fruit 8 40 That gives us a spread of 2.702 in Viet SS to sketch the shape of the sampling distribution with 12 and three standard deviations on either side of them. We're gonna do this. So we have a normal curve we have are mean, which is 280 and we're gonna put 12 and three standard deviations. And he said so one to the right would be approximately 2 82.1 to keep going to 84 0.2 one more to 86.3 and to the left of the mean be approximately 2 77.9 to 75.8 and finally to 73.7 in si says, According to the 68 95 99 7 rule, about 95% of all values of X bar lie within a distance. Film of the mean of the simply distribution wants to know what is him and then shade the region on the axis of the sketch that is within him of the mean. So in this case, um, we're dealing with the 95% of value, so the 95% value is gonna represent the two standard deviations. So to center. Deviations outside the mean on either side will put us at this to 75.8. Put us at to 84.2. This is the 95% region. According to the empirical rule of the 68 95 99.7, the ultimate loses Looking for him. Um um, which distance between the mean? So this essentially two standard deviations. So to center deviations in this case on either side is approximately 4.2. So we're going to say that I am is approximately 4.2, and in the region of shaded the boat and then Part D says, whenever explore falls in the region, you shaded the population mean utilizing a confidence interval X bar plus from honest for what percent of all possible values does the interval capture mu. So this is the idea of a confidence interval. So we're doing with this 95%. Um, we know that about 95% of the time we will capture the true I mean, if we're creating these complex intervals. So the whole idea is if we're trying to create multiple intervals using different samples 95% of the time, we will capture that true value within our confidence interval