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Chapter 9: Problem 5

A research firm conducted a survey to determine the mean amount steady smokers spend on cigarettes during a week. A sample of 49 steady smokers revealed that \(\bar{X}=\$ 20\) and \(s=\$ 5\) a. What is the point estimate of the population mean? Explain what it indicates. b. Using the 95 percent level of confidence, determine the confidence interval for \(\mu .\) Explain what it indicates.

Short Answer

Expert verified
a) $20; b) $18.60 to $21.40, 95% confidence mean is between these values.

Step by step solution

01

Identify the Point Estimate of the Population Mean

The point estimate of the population mean is the sample mean, which is denoted as \( \bar{X} \). In this problem, \( \bar{X} \) is given as \( \\(20 \). This indicates that the best estimate for the weekly spending of all steady smokers in the population, based on this sample, is \\)20.
02

Determine the Standard Error of the Mean

To calculate the confidence interval, start by finding the standard error of the mean (SEM). The formula for SEM is \( \frac{s}{\sqrt{n}} \), where \( s = \\(5 \) is the sample standard deviation and \( n = 49 \) is the sample size. Thus, \( SEM = \frac{5}{\sqrt{49}} = \frac{5}{7} = \\)0.7143 \).
03

Find the Critical Value for 95% Confidence Level

For a 95% confidence interval and a normal distribution, the critical value (\( z \)) is approximately 1.96. This value represents the number of standard deviations from the mean to the endpoints of the distribution for a 95% confidence level.
04

Calculate the Confidence Interval

Use the formula for the confidence interval: \( \bar{X} \pm z \times SEM \). Substituting the values, we have \( 20 \pm 1.96 \times 0.7143 \). Calculate the margin of error: \( 1.96 \times 0.7143 = 1.400 \). Thus, the confidence interval is \( 20 - 1.400 \) to \( 20 + 1.400 \), which simplifies to \( \\(18.60 \) to \( \\)21.40 \).
05

Interpret the Confidence Interval

The 95% confidence interval of \( \\(18.60 \) to \( \\)21.40 \) indicates that we can be 95% confident the true mean weekly spending on cigarettes by all steady smokers is between \\(18.60 and \\)21.40.

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Key Concepts

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

Population Mean
The population mean is a key concept in statistics. It represents the average value for an entire population. Imagine asking every steady smoker how much they spend on cigarettes weekly and averaging all these amounts. That's the population mean.
However, surveying everyone is often impractical. So, we use a smaller group, known as a sample, to get an idea of what the population mean might be.
In our scenario, we surveyed 49 steady smokers and their weekly spending. We want to use this information to estimate the overall average for all smokers. This estimation process revolves around our next concept, the point estimate.
Point Estimate
A point estimate provides a single value as a best guess for a population parameter, like the mean. In our case, this is the sample mean, represented as \( \ar{X} \).
For the steady smokers surveyed, the point estimate of their weekly spending on cigarettes is \( \\(20 \).
  • This means the average from our sample is \( \\)20 \).
  • It serves as our best prediction for the entire population's mean spending based on this sample data.
The point estimate is crucial because it gives us a concrete number to work with when predicting larger population behaviors. But to understand how accurate this estimate might be, we need to look at another important factor: the confidence interval.
Standard Error of the Mean
The standard error of the mean (SEM) helps us understand the precision of our sample mean as an estimate of the population mean.
It's calculated using the formula: \( \frac{s}{\sqrt{n}} \), where \( s \) is the sample's standard deviation, and \( n \) is the number of observations in the sample.
For example, with a standard deviation of \( \\(5 \) and a sample size of 49 smokers, the SEM is \( 0.7143 \).
  • This value tells us how much our sample mean \( (ar{X} = \\)20) \) is expected to vary from the true population mean on average.
  • A smaller SEM indicates a more precise estimate of the population mean.
Ultimately, the SEM is a cornerstone in constructing a confidence interval, offering insight into the reliability of our point estimate.

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