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Chapter 6: Problem 7

Mobile Phones in India India has over 600 million mobile phone subscribers. The largest company providing mobile phone service is Bharti Airtel, which has \(30 \%\) of the market share. \({ }^{2}\) If random samples of 500 mobile phone subscribers in India are selected and we compute the proportion using service from Bharti Airtel, find the mean and the standard error of the sample proportions.

Short Answer

Expert verified
The mean of the sample proportions is 0.30 and the standard error is 0.02048.

Step by step solution

01

Find the Population Proportion

The proportion (P) of subscribers using service from Bharti Airtel in the population is given as \(30 \%\) or 0.30.
02

Calculate the Mean of Sample Proportions

We know, the mean (\(\mu\)) of any sample proportions is always equal to the population proportion (P). Thus, the mean (\(\mu\)) of sample proportions will also be 0.30.
03

Find the Standard Deviation of the Population Proportion

The standard deviation (\(\sigma\)) of the population proportion is calculated as \(\sigma = \sqrt{P*(1-P)} = \sqrt{0.30*(1-0.30)} = \sqrt{0.21} = 0.45826.\)
04

Determine the Standard Error of the Sample Proportions

The formula for standard error (SE) is \(\sigma/\sqrt{n}\), where 'n' is the sample size. Substituting the values, the standard error (SE) is calculated as \(0.45826/\sqrt{500} = 0.02048.\)

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Key Concepts

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

Population Proportion
When dealing with statistics, it's essential to understand what the term **Population Proportion** means. This concept refers to the fraction of the total population that exhibits a particular characteristic. In our exercise, the population proportion is the percentage of Indian mobile phone users who subscribe to Bharti Airtel. This is given as \( 30\% \) or 0.30.
  • In simple terms, if we consider the entire population of mobile phone users in India, 30 out of every 100 users are using Airtel services.
  • The population proportion serves as a reference point or a baseline when we take and analyze samples from this population.
Understanding the population proportion is crucial as it forms the basis for calculating other statistical measures like the mean of sample proportions and standard error. By knowing that the proportion is 0.30, we can use it to determine other related statistics for sample data.
Standard Error
In statistics, the **Standard Error** tells us how much variability or "spread" we can expect when we take multiple samples from a population. It shows us how much the sample proportion is likely to vary from the actual population proportion.
  • The standard error of the sample proportion is calculated using the formula \( SE = \sigma/\sqrt{n} \), where \( \sigma \) is the standard deviation of the population proportion, and \( n \) is the sample size.
  • For our example, we already calculated the standard deviation as 0.45826. The sample size is 500, so the standard error comes out to \( 0.02048 \).
  • This value tells us that when we take samples of 500 mobile users, their sample proportion will generally deviate by about 0.02048 from the population proportion.
The smaller the standard error, the more precise our sample proportion is likely to be in representing the actual population proportion.
Mean of Sample Proportions
The **Mean of Sample Proportions** is an important concept when drawing conclusions from sample data. Often denoted as \( \mu \), it is the average of sample proportions over infinite samples and equals the population proportion. In our exercise, since the population proportion of Bharti Airtel users is 0.30, the mean of the sample proportions is also 0.30.
  • In essence, if we were to repeatedly draw samples and calculate the proportion of Airtel subscribers, the average of these proportions would be 0.30.
  • This relationship is an excellent property of random samples; it assures us that despite variations among samples, they center around the true population proportion.
Being aware of the mean of sample proportions provides confidence in the reliability of samples in estimating population proportions over time.

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Most popular questions from this chapter

We saw in Exercise 6.260 on page 425 that drinking tea appears to offer a strong boost to the immune system. In a study extending the results of the study described in that exercise, \(^{70}\) blood samples were taken on five participants before and after one week of drinking about five cups of tea a day (the participants did not drink tea before the study started). The before and after blood samples were exposed to e.coli bacteria, and production of interferon gamma, a molecule that fights bacteria, viruses, and tumors, was measured. Mean production went from 155 \(\mathrm{pg} / \mathrm{mL}\) before tea drinking to \(448 \mathrm{pg} / \mathrm{mL}\) after tea drinking. The mean difference for the five subjects is \(293 \mathrm{pg} / \mathrm{mL}\) with a standard deviation in the differences of 242 . The paper implies that the use of the t-distribution is appropriate. (a) Why is it appropriate to use paired data in this analysis? (b) Find and interpret a \(90 \%\) confidence interval for the mean increase in production of interferon gamma after drinking tea for one week.

In Exercises 6.166 and \(6.167,\) use StatKey or other technology to generate a bootstrap distribution of sample differences in proportions and find the standard error for that distribution. Compare the result to the value obtained using the formula for the standard error of a difference in proportions from this section (and using the sample proportions to estimate the population proportions). Sample A has a count of 30 successes with \(n=100\) and Sample \(\mathrm{B}\) has a count of 50 successes with \(n=250\).

The All Countries dataset includes land area, in square kilometers, for all 213 countries in the world. The mean land area for all the countries is \(608,120 \mathrm{sq} \mathrm{km}\) with standard deviation 1,766,860 . For samples of size \(50,\) what percentage of sample means will be less than 400,000 sq km? What percentage will be greater than \(900,000 ?\)

In Exercises 6.1 to \(6.6,\) if random samples of the given size are drawn from a population with the given proportion: (a) Find the mean and standard error of the distribution of sample proportions. (b) If the sample size is large enough for the Central Limit Theorem to apply, draw a curve showing the shape of the sampling distribution. Include at least three values on the horizontal axis. Samples of size 50 from a population with proportion 0.25

Exercise 4.86 on page 263 introduces a matched pairs study in which 47 participants had cell phones put on their ears and then had their brain glucose metabolism (a measure of brain activity) measured under two conditions: with one cell phone turned on for 50 minutes (the "on" condition) and with both cell phones off (the "off" condition). Brain glucose metabolism is measured in \(\mu \mathrm{mol} / 100 \mathrm{~g}\) per minute, and the differences of the metabolism rate in the on condition minus the metabolism rate in the off condition were computed for all participants. The mean of the differences was 2.4 with a standard deviation of \(6.3 .\) Find and interpret a \(95 \%\) confidence interval for the effect size of the cell phone waves on mean brain metabolism rate.
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