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Chapter 5: Q63SE (page 323)

Improving SAT scores. Refer to the Chance(Winter2001) examination of Scholastic Assessment Test (SAT)scores of students who pay a private tutor to help them improve their results, Exercise 2.88 (p. 113). On the SAT—Mathematics test, these students had a mean score change of +19 points, with a standard deviation of 65 points. In a random sample of 100 students who pay a private tutor to help them improve their results, what is the likelihood that the sample mean score change is less than 10 points?

Short Answer

Expert verified

The likelihood that the sample means score change is less than 10 points is 0.0838.

Step by step solution

01

Given information

Referring to exercise 2.88 (p.113), there is a study about 100 students who pay a private tutor to help them improve their results. The students had a mean score of 19 and a standard deviation of 65 points.

02

Determine the likelihood

Let’s consider the sample mean of μx¯=19.

And the sample standard deviation of

σx¯=σn=65100=6.5

So, the likelihood that the sample mean score change is less than 10 points is,

PrX¯<10=PrX¯-μx¯σx¯<10-196.5=Prz<-1.38=0.0838

Therefore, the likelihood is 0.0838.

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Most popular questions from this chapter

Question: The standard deviation (or, as it is usually called, the standard error) of the sampling distribution for the sample mean, x¯ , is equal to the standard deviation of the population from which the sample was selected, divided by the square root of the sample size. That is

σX¯=σn

  1. As the sample size is increased, what happens to the standard error of? Why is this property considered important?
  2. Suppose a sample statistic has a standard error that is not a function of the sample size. In other words, the standard error remains constant as n changes. What would this imply about the statistic as an estimator of a population parameter?
  3. Suppose another unbiased estimator (call it A) of the population mean is a sample statistic with a standard error equal to

σA=σn3

Which of the sample statistics,x¯or A, is preferable as an estimator of the population mean? Why?

  1. Suppose that the population standard deviation σis equal to 10 and that the sample size is 64. Calculate the standard errors of x¯and A. Assuming that the sampling distribution of A is approximately normal, interpret the standard errors. Why is the assumption of (approximate) normality unnecessary for the sampling distribution ofx¯?

Corporate sustainability of CPA firms. Refer to the Business and Society (March 2011) study on the sustainability behaviours of CPA corporations, Exercise 1.28 (p. 51). Corporate sustainability, recall, refers to business practices designed around social and environmental considerations. The level of support senior managers has for corporate sustainability was measured quantitatively on a scale ranging from 0 to 160 points. The study provided the following information on the distribution of levels of support for sustainability:μ=68 , σ=27. Now consider a random sample of 45 senior managers and let x represent the sample mean level of support.

a. Give the value of μx¯, the mean of the sampling distribution ofx¯ , and interpret the result.

b. Give the value ofσx¯ , the standard deviation of the sampling distribution of x¯, and interpret the result.

c. What does the Central Limit Theorem say about the shape of the sampling distribution ofx¯ ?

d. Find Px¯>65.

Do social robots walk or roll? Refer to the International Conference on Social Robotics (Vol. 6414, 2010) study of the trend in the design of social robots, Exercise 2.5 (p. 72). The researchers obtained a random sample of 106 social robots through a Web search and determined the number that was designed with legs but no wheels. Let p^represent the sample proportion of social robots designed with legs but no wheels. Assume that in the population of all social robots, 40% are designed with legs but no wheels.

a. Give the mean and standard deviation of the sampling distribution of p^.

b. Describe the shape of the sampling distribution of p^.

c. Find P(p^>.59).

d. Recall that the researchers found that 63 of the 106 robots were built with legs only. Does this result cast doubt on the assumption that 40% of all social robots are designed with legs but no wheels? Explain.

A random sample of n= 300 observations is selectedfrom a binomial population with p= .8. Approximateeach of the following probabilities:

  1. Pp^<0.83
  2. Pp^>0.75
  3. P0.79<p^<0.81

Refer to Exercise 5.3.

  1. Show thatx⇀is an unbiased estimator of.
  2. Findσx2.
  3. Find the probability that x will fall within2σxofμ.
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