Question

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, 32 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

The population of interest is all US cell phone users aged 18 and older, and the parameter of interest is the average number of text messages sent or received per day. Using the data from the sample of 755 users, the best estimate of the population parameter is 41.5 messages. The 95% confidence interval is calculated as roughly [29.74, 53.26]. That means we are 95% confident that the true average number of text messages sent or received per day by all US cell phone users aged 18 and older is within approximately 29.74 and 53.26.

Step-by-step solution

Identify the population and parameter of the interest

The population of interest refers to all the people or items that the statistics are about. In this case, the population consists of all US cell phone users aged 18 and older. The parameter of interest is the average number of text messages sent or received per day. The best estimate of the population parameter is the sample mean, which here amounts to 41.5 messages.

Calculation of the Confidence Interval

The formula of a confidence interval is \(\widehat{p} \pm Z \cdot \sqrt{\frac{\widehat{p} (1 - \widehat{p})}{n}}\) where \(\widehat{p}\) is the sample proportion, \(Z\) is the z-value (for a 95% confidence interval, this value is 1.96), and \(n\) is the sample size. Using the data gathered, the confidence interval is calculated as 41.5 ± (1.96 × 6.1).

Interpretation of the Confidence Interval

Once the lower and upper bounds of the confidence interval are calculated, these need to be interpreted in the context of the problem. For instance, based on the calculations, we are 95% confident that the true number of text messages sent or received per day by all US cell phone users aged 18 and older falls within this interval.

Key concepts

Population Parameter

When we talk about the population parameter, we're referring to a specific characteristic or measure of an entire group. For instance, the average height of all adults in a city, the average temperature of a region over a year, or, as in our exercise, the average number of text messages sent or received per day by all US cell phone users aged 18 and older.

Understanding the population parameter is crucial because it gives us insight into the entire group we are studying without having to check every single member. However, it's often impractical or impossible to measure the population parameter directly due to the large number of individuals. Therefore, statisticians use a sample—a manageable size group representing the larger population—to estimate the population parameter.

In the given exercise, the average number of text messages sent or received per day is estimated using a sample mean, which serves as the best guess for the population parameter. We assume this sample is representative of the population, an assumption that underlies much of statistical inference.

Sample Mean

The sample mean is the arithmetic average of all measurements in a sample. It's used as an estimate of the population mean (the average of the entire population). In statistical terms, if the sample is drawn randomly and is sufficiently large, the sample mean tends to be a good estimator of the population mean.

In our text messaging exercise, a sample of 755 US cell phone users gives a sample mean of 41.5 messages per day. It’s important to remember that because samples vary, sample means will vary as well. This variation gives rise to the concept of the standard error which measures how much the sample mean is expected to fluctuate due to random sampling variability. The standard error in the exercise is 6.1, implying that if different samples were taken, the sample means would typically differ from the true mean by about 6.1 messages per day.

To improve comprehension, students should visualize these concepts. Picture the population as a vast ocean, and the sample mean as an average depth determined by several measurements at different locations. The standard error is like the variation in depth between these points; it’s usually not the same throughout the ocean.

Confidence Level

The confidence level is a measure of certainty - or confidence - that the population parameter falls within the confidence interval calculated from the sample. Common confidence levels include 90%, 95%, and 99%. In the text message exercise, the confidence level used is 95%. This means that if we were to take many different samples and calculate the confidence interval for each one, about 95% of these intervals would contain the true population mean.

Here's a simpler way to think about it: If you were placing bets that a friend will catch a bus that arrives every 10 minutes, a 95% confidence level is like saying, 'I'm 95% sure you'll catch a bus within 20 minutes.' Statistically speaking, there's a small chance you could be wrong, but it's a strong bet.

In the context of our exercise, we calculated the confidence interval as 41.5 ± (1.96 × 6.1). The interval captures the range within which we can say, with 95% certainty, that the true mean lies. If this concept still seems hazy, picture drawing a circle around a target; you're confident that most arrows will hit within this circle, but there's still a chance some might miss, representing the 5% uncertainty.