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Chapter 17: Problem 22

Pew Research reported that, in \(2013,78 \%\) of all teens had a cell phone. Assume this estimate is correct. a) We randomly pick 100 teens. Let \(\hat{p}\) represent the proportion of teens in this sample who own a cell phone. What's the appropriate model for the distribution of \(\hat{p} ?\) Specify the name of the distribution, the mean, and the standard deviation. Be sure to verify that the conditions are met. b) What's the approximate probability that less than three fourths of this sample own a cell phone?

Short Answer

Expert verified
The distribution model for \( \hat{p} \) is normal with mean 0.78 and standard deviation 0.042. The probability that less than 75% own a cell phone is about 23.8%.

Step by step solution

01

Understand the problem

We are given that 78% of all teens own a cell phone, which will be the probability of success (p) in our sample. We need to determine the distribution describing the proportion of teens in a sample who own a cell phone, then compute the probability that less than 75% of them own a cell phone.
02

Identify the distribution model

For a sample proportion \( \hat{p} \), the distribution can be modeled using a normal distribution if the sample size is large enough. This is called the sampling distribution of the sample proportion. It applies here because the sample size is 100, which is generally considered adequate for normal approximation.
03

Verify conditions for normal approximation

Check two key conditions: \(np \geq 10\) and \(n(1-p) \geq 10\). Here, \(n = 100\) and \(p = 0.78\). Hence, \( np = 100 \times 0.78 = 78 \) and \(n(1-p) = 100 \times 0.22 = 22\). Both conditions are satisfied, allowing us to use the normal distribution.
04

Determine mean and standard deviation

For the sample proportion \( \hat{p} \), the mean \( \mu_{\hat{p}} \) is equal to \( p \), so \( \mu_{\hat{p}} = 0.78 \). The standard deviation \( \sigma_{\hat{p}} \) is given by \( \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.78 \times 0.22}{100}} = 0.042\).
05

Calculate the probability

We seek the probability that \( \hat{p} < 0.75 \). First, find the z-score: \( z = \frac{0.75 - 0.78}{0.042} = -0.71\). Using a standard normal distribution table or calculator, find \( P(Z < -0.71) \), which approximately equals 0.238. Therefore, there is a 23.8% chance that less than 75% of the sample own a cell phone.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution
A normal distribution is a continuous probability distribution characterized by its bell-shaped curve. This distribution is symmetric around the mean, meaning that data near the mean are more frequent in occurrence than data far from the mean.
It's a fundamental concept in statistics as it describes how the values of a variable are distributed. In the context of sampling distribution, when sample sizes are large, the distribution of the sample mean approaches a normal distribution. This is thanks to the Central Limit Theorem.
For the exercise at hand, the normal distribution is used to model the sampling distribution of the sample proportion, since the sample size of 100 is large enough. This allows us to make probability inferences about the sample proportion using the properties of the normal distribution.
Sample Proportion
The sample proportion, denoted as \( \hat{p} \), is the ratio of individuals in a sample with a particular characteristic to the total number of individuals in the sample. It is a statistic that estimates the population proportion \( p \).
In our example, \( \hat{p} \) refers to the proportion of teens in a sample of 100 who own a cell phone.
The sample proportion follows a special distribution known as the sampling distribution of the sample proportion. When the sample size is large enough, this distribution can be approximated by a normal distribution with a mean \( p \) and a standard deviation given by \( \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \).
This strength of approximation forms the basis of conducting probability calculations on sample proportions.
Probability Calculation
Probability calculation involves determining the likelihood of an event occurring. In the statistical context, especially when dealing with sampling distribution, it often involves the normal distribution.
In this exercise, we calculate the probability that less than 75% of the teens in the sample own a cell phone. To find this probability, we first determine the Z-score for the sample proportion of 0.75 and then use a standard normal distribution table or calculator to find the corresponding probability.
The calculated probability, in this case, describes the likelihood of observing a sample proportion that's less than the specified 0.75 threshold, given the population proportion of 0.78.
Z-Score
The Z-score is a crucial metric in statistics that indicates how many standard deviations an element is from the mean of the distribution. It is calculated using the formula \( z = \frac{X - \mu}{\sigma} \), where \( X \) is the value of the sample statistic, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
In the context of our exercise, the Z-score calculation helps us determine how far the desired sample proportion (75%) is from the mean of the sampling distribution (which is 78%).
A negative Z-score, as calculated in this exercise (-0.71), indicates that the sample proportion is less than the population proportion. By using the Z-score and referring to the standard normal distribution table, you can assess the probability or likelihood of obtaining various sample proportions, allowing us to determine that there's a 23.8% probability that less than 75% of teens in the sample own a cell phone.

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Most popular questions from this chapter

Based on past experience, a bank believes that \(7 \%\) of the people who receive loans will not make payments on time. The bank has recently approved 200 loans. a) What are the mean and standard deviation of the proportion of clients in this group who may not make timely payments? b) What assumptions underlie your model? Are the conditions met? Explain. c) What's the probability that over \(10 \%\) of these clients will not make timely payments?

The weight of potato chips in a medium size bag is stated to be 10 ounces. The amount that the packaging machine puts in these bags is believed to have a Normal model with mean 10.2 ounces and standard deviation 0.12 ounces. a) What fraction of all bags sold are underweight? b) Some of the chips are sold in "bargain packs" of 3 bags. What's the probability that none of the 3 is underweight? c) What's the probability that the mean weight of the 3 bags is below the stated amount? d) What's the probability that the mean weight of a 24 -bag case of potato chips is below 10 ounces?

The candy company claims that \(16 \%\) of the Milk Chocolate M\&M's it produces are green. Suppose that the candies are thoroughly mixed and then packaged in small bags containing about \(50 \mathrm{M} \& \mathrm{M}^{\prime} \mathrm{s}\). A class of elementary school students learning about percents opens several bags, counts the various colors of the candies, and calculates the proportion that are green. a) If we plot a histogram showing the proportions of green candies in the various bags, what shape would you expect it to have? b) Can that histogram be approximated by a Normal model? Explain. c) Where should the center of the histogram be? d) What should the standard deviation of the sampling distribution be?

Rainfall Statistics from Cornell's Northeast Regional Climate Center indicate that Ithaca, NY, gets an average of \(35.4^{\prime \prime}\) of rain each year, with a standard deviation of \(4.2^{\prime \prime} .\) Assume that a Normal model applies. a) During what percentage of years does Ithaca get more than \(40^{\prime \prime}\) of rain? b) Less than how much rain falls in the driest \(20 \%\) of all years? c) A Cornell University student is in Ithaca for 4 years. Let \(\bar{y}\) represent the mean amount of rain for those 4 years. Describe the sampling distribution model of this sample mean, \(\bar{y}\) d) What's the probability that those 4 years average less than \(30^{\prime \prime}\) of rain?

According to a Pew Research survey, about \(27 \%\) of American adults are pessimistic about the future of marriage and the family. This is based on a sample, but assume that this percentage is correct for all American adults. a) Using a binomial model, what is the probability that, in a sample of 20 American adults, \(25 \%\) or fewer of the people in the sample are pessimistic about the future of marriage and family? b) Now use a Normal model to compute that probability. How does this compare to your answer from part a? c) Using a Binomial model, what is the probability that, in a sample of 700 American adults, \(25 \%\) or fewer of the people of the people in the sample are pessimistic about the future of marriage and family? d) Now use a Normal model to compute that probability. How does this compare to your answer from part c? e) What do these answers tell you about the importance of checking that \(n p\) and \(n q\) are both at least \(10 ?\)
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