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Chapter 3: Problem 54

3.54 Number of Text Messages a Day A random sample of \(n=755\) US cell phone users age 18 and older in May 2011 found that the average number of text messages sent or received per day is 41.5 messages, \({ }^{25}\) with standard error about 6.1 (a) State the population and parameter of interest. Use the information from the sample to give the best estimate of the population parameter. (b) Find and interpret a \(95 \%\) confidence interval for the mean number of text messages.

Short Answer

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a) The population consists of all US cell phone users aged 18 and older, and the parameter of interest is the average number of text messages they send per day. The best estimate for this parameter is 41.5 messages/day (the sample mean). b) The 95% confidence interval for the mean number of messages sent/received per day is between 29.5 and 53.5. We interpret this as follows: We are 95% confident that the population mean lies in the interval 29.5 to 53.5 messages/day.

Step by step solution

01

Define the Population and Parameter

The population that we are interested in is all the cell phone users in the US aged 18 and older. The parameter of interest is the average number of messages sent or received per day by these users. The best estimate of this population parameter is given by the sample mean, which is 41.5 messages per day.
02

Finding the Standard Error

The standard error (se) of the sample mean is given as 6.1. It gives us the standard deviation of the distribution of the sample mean.
03

Calculate the 95% Confidence Interval

A 95% confidence interval for the population mean (\(渭\)) can be calculated by the following two formulas, \(\overline{x}-Z脳se\) and \(\overline{x}+Z脳se\), where \(\overline{x}\) is the sample mean, \(Z\) is the critical value of the standard normal distribution corresponding to the desired confidence level (1.96 for 95% confidence), and \(se\) is the standard error. So, the lower limit of the confidence interval would be \(41.5-1.96脳6.1 鈮� 29.5\), and the upper limit would be \(41.5+1.96脳6.1 鈮� 53.5\).
04

Interpret the Confidence Interval

The interpretation of the confidence interval is that we are 95% confident that the average number of messages sent or received per day by US cell phone users aged 18 and older is between 29.5 and 53.5.

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Key Concepts

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

Standard Error
Standard error is a key concept in statistics that measures the accuracy with which a sample mean presents, or estimates, a population mean. It is derived from the standard deviation of the sample mean distribution.
  • The standard error is useful because it reflects how much the sample mean is expected to fluctuate around the true population mean.
  • A lower standard error suggests that the sample mean is a more accurate estimate of the population mean.
The formula for calculating the standard error is \[ se = \frac{s}{\sqrt{n}} \]where:
  • \( s \) is the standard deviation of the sample.
  • \( n \) is the sample size.
In the problem, the standard error is given as 6.1. This means that the sample mean of 41.5 messages per day fluctuates around the true population mean in a predictable way by roughly 6.1 messages, showing how spread out the sample means are likely to be.
Sample Mean
The sample mean is the average value of a data set, calculated from a sample of a larger population. It is often used to estimate the population mean, representing the central tendency of a dataset.
  • For example, in the exercise, the sample mean is 41.5 messages per day. This value is obtained from a random sample of 755 US cell phone users aged 18 and older.
  • The sample mean helps us understand what the average cell phone user might communicate through text messages on a daily basis.
Sample means are crucial because they provide an estimate of the population mean, even if it is impractical to measure every individual in the population. This estimation leads to insights into the population's behavior based on just the sample.
Population Parameter
The population parameter is a numerically valued attribute of a model that describes a characteristic of a population. It reflects certain properties of the entire population under study.
  • In this context, the population parameter of interest is the average number of text messages sent or received by US cell phone users aged 18 and older.
  • This is an unknown value, but one can use the sample statistics to estimate it 鈥� the sample mean being 41.5 messages provides this estimation.
Unlike the sample statistics that vary from sample to sample, the population parameter remains constant. Accurate estimation of a population parameter allows us to grasp a clear understanding of the population's characteristics, ensuring informed decision-making in various fields such as market research, policy-making, and business strategy.

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Most popular questions from this chapter

How Important Is Regular Exercise? In a recent poll \(^{42}\) of 1000 American adults, the number saying that exercise is an important part of daily life was \(753 .\) Use StatKey or other technology to find and interpret a \(90 \%\) confidence interval for the proportion of American adults who think exercise is an important part of daily life.

Daily Tip Revenue for a Waitress Data 2.12 on page 119 describes information from a sample of 157 restaurant bills collected at the First Crush bistro. The data is available in RestaurantTips. Two intervals are given below for the average tip left at a restaurant; one is a \(90 \%\) confidence interval and one is a \(99 \%\) confidence interval. Interval A: 3.55 to \(4.15 \quad\) Interval B: 3.35 to 4.35 (a) Which one is the \(90 \%\) confidence interval? Which one is the \(99 \%\) confidence interval? (b) One waitress generally waits on 20 tables in an average shift. Give a range for her expected daily tip revenue, using both \(90 \%\) and \(99 \%\) confidence. Interpret your results.

A \(95 \%\) confidence interval is given, followed by possible values of the population parameter. Indicate which of the values are plausible values for the parameter and which are not. \(\mathbf{3 . 4 4}\) A \(95 \%\) confidence interval for a proportion is 0.72 to \(0.79 .\) Is the value given a plausible value of \(p ?\) (a) \(p=0.85\) (b) \(p=0.75\) (c) \(p=0.07\)

Have You Ever Been Arrested? According to a recent study of 7335 young people in the US, \(30 \%\) had been arrested \(^{21}\) for a crime other than a traffic violation by the age of \(23 .\) Crimes included such things as vandalism, underage drinking, drunken driving, shoplifting, and drug possession. (a) Is the \(30 \%\) a parameter or a statistic? Use the correct notation. (b) Use the information given to estimate a parameter, and clearly define the parameter being estimated. (c) The margin of error for the estimate in part (b) is 0.01 . Use this information to give a range of plausible values for the parameter. (d) Given the margin of error in part (c), if we asked all young people in the US if they have ever been arrested, is it likely that the actual proportion is less than \(25 \% ?\)

Florida has over 7700 lakes. \(^{11}\) We wish to estimate the correlation between the \(\mathrm{pH}\) levels of all Florida lakes and the mercury levels of fish in the lakes. We see in Data 2.4 on page 68 that the correlation between these two variables for a sample of \(n=53\) of the lakes is -0.575 . (a) Give notation for the quantity we are estimating, notation for the quantity we use to make the estimate, and the value of the best estimate. (b) Why is an estimate necessary here? What would we have to do to calculate the exact value of the quantity we are estimating?
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