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Chapter 8: Q 59. (page 524)

Oranges A home gardener likes to grow various kinds of citrus fruit. One of his

mandarin orange trees produces oranges whose circumferences follow a Normal

distribution with mean 21.1cm and standard deviation 1.8cm.

a. What is the probability that a randomly selected orange from this tree has a

circumference greater than 22cm?

b. What is the probability that a random sample of 20 oranges from this tree has a mean circumference greater than 22 cm?

Short Answer

Expert verified

a. The probability is 0.3085.

b. The probability is0.012676

Step by step solution

01

Given Information

It is given that (μ)=21.1

(σ)=1.8

n=20

02

Probability that a randomly selected orange from this tree has acircumference greater than 22 cm

Let Xbe a random variable representing circumference of orange following normal distribution having mean as 21.1cm

Standard deviation 1.8cm

Required probability

P(X>22)=Px-μσ>22-21.11.8

=P(Z>0.5)

=1-P(Z<0.5)

=0.3085

03

Probability that a random sample of 20 oranges from this tree has a mean circumference greater than 22cm

Required probability:

P(X¯>22)=Px¯-μσn>22-21.11.820

=P(Z>2.236)

=1-P(Z<2.236)

=0.012676

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estimated that 49% of U.S. adults play video games on a computer, TV, game console, or portable device such as a cell phone. This estimate was based on a random sample of 2001 U.S. adults. Construct and interpret a 95% confidence interval for the proportion of all U.S. adults who play video games.

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Explaining confidence The admissions director for a university found that (107.8,116.2)is a 95%confidence interval for the mean IQ score of all freshmen. Discuss whether each of the following explanations is correct, based on that information.

a. There is a 95%probability that the interval from role="math" localid="1654200953396" 107.8to116.2contains μ.

b. There is a 95%chance that the interval(107.8,116.2)contains x¯.

c. This interval was constructed using a method that produces intervals that capture the true mean in 95%of all possible samples.

d. If we take many samples, about 95%of them will contain the interval (107.8,116.2).

e. The probability that the interval (107.8,116.2)captures μ is either 0 or 1, but we don’t know which.
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