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Chapter 7: Problem 71

In a large city, \(88 \%\) of the cases of car burglar alarms that go off are false. Let \(\hat{p}\) be the proportion of false alarms in a random sample of 80 cases of car burglar alarms that go off in this city. Calculate the mean and standard deviation of \(\hat{p}\), and describe the shape of its sampling distribution.

Short Answer

Expert verified
The mean of the proportion \( \hat{p} \) is 0.88, its standard deviation is calculated from the given formula to be \( \sqrt{\frac{0.88(1-0.88)}{80}} \) and the distribution is approximately normally distributed.

Step by step solution

01

Calculation of Mean

The mean of the sampling distribution of the proportion, often denoted as \( \mu_{\hat{p}} \) is simply the value of the proportion in the population, given as \( p \). In this case, the problem has already stated that the proportion of false alarms (\( p \)) is \( 0.88 \). So, \( \mu_{\hat{p}} = 0.88 \).
02

Calculation of Standard Deviation

The formula for the standard deviation of the sampling distribution of a proportion (\( \sigma_{\hat{p}} \)) is given by \( \sqrt{ \frac{p(1-p)}{n} } \), where \( p \) is the population proportion and \( n \) is the sample size. In this case, \( p = 0.88 \) and \( n = 80 \). Hence, \( \sigma_{\hat{p}} = \sqrt{\frac{0.88(1-0.88)}{80}} \).
03

Shape of The Sampling Distribution

The shape of the sampling distribution can be determined by the rule of thumb which states that if both \( np \) and \( n(1-p) \) are greater than 5 then the proportion distribution is approximately normally distributed. Here, \( np = 80*0.88 = 70.4 \) and \( n(1-p) = 80*(1-0.88)=9.6 \). Both are greater than 5, hence the distribution is approximately normally distributed.

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Most popular questions from this chapter

The amounts of electricity bills for all households in a particular city have an approximate normal distribution with a mean of \(\$ 140\) and a standard deviation of \(\$ 30\). Let \(\bar{x}\) be the mean amount of electricity bills for a random sample of 25 households selected from this city. Find the mean and standard deviation of \(\bar{x}\), and comment on the shape of its sampling distribution.

The package of Ecosmart Led 75-watt replacement bulbs that use only 14 watts claims that these bulbs have an average life of 24,966 hours. Assume that the lives of all such bulbs have an approximate normal distribution with a mean of 24,966 hours and a standard deviation of 2000 hours. Find the probability that the mean life of a random sample of 25 such bulbs is a. less than 24,400 hours b. between 24,300 and 24,700 hours c. within 650 hours of the population mean d. less than the population mean by 700 hours or more

Indicate in which of the following cases the central limit theorem will apply to describe the sampling distribution of the sample proportion. a. \(n=400\) and \(p=.28\) b. \(n=80\) and \(p=.05\) c. \(n=60\) and \(p=.12\) d. \(n=100\) and \(p=.035\)

Let \(x\) be a continuous random variable that has a normal distribution with \(\mu=48\) and \(\sigma=8\). Assuming \(n / N \leq .05\), find the probability that the sample mean, \(\bar{x}\), for a random sample of 16 taken from this population will be a. between \(49.6\) and \(52.2\) b. more than \(45.7\)

A survey of all medium- and large-sized corporations showed that \(64 \%\) of them offer retirement plans to their employees. Let \(\hat{p}\) be the proportion in a random sample of 50 such corporations that offer retirement plans to their employees. Find the probability that the value of \(\hat{p}\) will be a. between \(.54\) and \(.61\) b. greater than \(.71\)
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