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Chapter 5: 5-Q38E (page 320)

A random sample of n = 80 measurements is drawn from a binomial population with a probability of success .3.

  1. Give the mean and standard deviation of the sampling distribution of the sample proportion,P^
  2. Describe the shape of the sampling distribution ofP铃�
  3. Calculate the standard normal z-score corresponding to a value ofP铃�=.35.
  4. FindP(P铃�=.35.)

Short Answer

Expert verified

A random sample is a subgroup of people chosen at random by investigators to represent the full population as a whole. A technique for selecting a sample of data from a community to make predictions regarding the community.

Step by step solution

01

(a) The data is given below

The calculation is given below:

Given,

n = 80p = 0.3

Meanp= 0.3蟽P铃�=pqn=0.3脳1-0.380=0.0512

鈮�0.05

02

(b) The data is given below

The calculation is given below:

np = 80脳0.3 = 24猢�10nq = 80脳0.7 = 56猢�10

Since np猢�10 and nq猢�10,the sample probability is normal distribution is normally distributed with 渭P铃�=0.3 and蟽P铃�= 0.05

03

(c) The data is given below

The calculation is given below:

ForP铃�=0.35, the Z score is:

Z=P铃�-渭P铃�蟽P铃�=0.35-0.30.05=1

04

(d) The data is given below

The calculation is given below:

PP铃�>.35=PP铃�-渭P铃�蟽P铃�猢�0.35-渭p铃�蟽p铃�=PZ猢�1=1-PZ<1

=1-0.8413=0.1587

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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炉 , 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

蟽X炉=蟽n

  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

蟽A=蟽n3

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

  1. Suppose that the population standard deviation 蟽is equal to 10 and that the sample size is 64. Calculate the standard errors of x炉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炉?

Refer to Exercise 5.18. Find the probability that

  1. x炉is less than 16.
  2. x炉is greater than 23.
  3. x炉is greater than 25.
  4. x炉falls between 16 and 22.
  5. x炉 is less than 14.

A random sample of n = 64 observations is drawn from a population with a mean equal to 20 and a standard deviation equal to 16

a. Give the mean and standard deviation of the (repeated) sampling distribution of x.

b. Describe the shape of the sampling distribution of x. Does your answer depend on the sample size?

c. Calculate the standard normal z-score corresponding to a value of x = 15.5.

d. Calculate the standard normal z-score corresponding to x = 23

Question: Hotel guest satisfaction. Refer to the results of the 2015 North American Hotel Guest Satisfaction Index Study, Exercise 4.49 (p. 239). Recall that 15% of hotel guests were 鈥渄elighted鈥� with their experience (giving a rating of 10 out of 10); of these guests, 80% stated they would 鈥渄efinitely鈥� recommend the hotel. In a random sample of 100 hotel guests, find the probability that fewer than 10 were delighted with their stay and would recommend the hotel.

Switching banks after a merger. Banks that merge with others to form 鈥渕ega-banks鈥� sometimes leave customers dissatisfied with the impersonal service. A poll by the Gallup Organization found 20% of retail customers switched banks after their banks merged with another. One year after the acquisition of First Fidelity by First Union, a random sample of 250 retail customers who had banked with First Fidelity were questioned. Letp^ be the proportion of those customers who switched their business from First Union to a different bank.

  1. Find the mean and the standard deviation of role="math" localid="1658320788143" p^.
  2. Calculate the interval Ep^卤2蟽p^.
  3. If samples of size 250 were drawn repeatedly a large number of times and determined for each sample, what proportion of the values would fall within the interval you calculated in part c?
See all solutions

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