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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

Average Penalty Minutes in the NHL In Exercise 3.86 on page \(204,\) we construct an interval estimate for mean penalty minutes given to NHL players in a season using data from players on the Ottawa Senators as our sample. Some percentiles from a bootstrap distribution of 1000 sample means are shown below. Use this information to find and interpret a \(98 \%\) confidence interval for the mean penalty minutes of NHL players. Assume that the players on this team are a reasonable sample from the population of all players.

Information about a sample is given. Assuming that the sampling distribution is symmetric and bell-shaped, use the information to give a \(95 \%\) confidence interval, and indicate the parameter being estimated. $$ r=0.34 \text { and the standard error is } 0.02 . $$

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What Proportion of Adults and Teens Text Message? A study of \(n=2252\) adults age 18 or older found that \(72 \%\) of the cell phone users send and receive text messages. \(^{14}\) A study of \(n=800\) teens age 12 to 17 found that \(87 \%\) of the teen cell phone users send and receive text messages. What is the best estimate for the difference in the proportion of cell phone users who use text messages, between adults (defined as 18 and over) and teens? Give notation (as a difference with a minus sign) for the quantity we are trying to estimate, notation for the quantity that gives the best estimate, and the value of the best estimate. Be sure to clearly define any parameters in the context of this situation.

Give the correct notation for the quantity described and give its value. Mean number of cell phone calls made or received per day by cell phone users. In a survey of 1917 cell phone users, the mean was 13.10 phone calls a day. \(^{8}\)
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