Question

A researcher has gathered information from a random sample of 178 households. For each of the following variables, construct confidence intervals to estimate the population mean. Use the \(90 \%\) level. a. An average of \(2.3\) people resides in each household. Standard deviation is \(0.35\). b. There was an average of \(2.1\) television sets \((s=0.10)\) and \(0.78\) telephones \((s=0.55)\) per household. c. The households averaged \(6.0\) hours of television viewing per day \((s=3.0)\)

Short answer

a. [2.257, 2.343], b. [2.088, 2.112] (TV sets) and [0.712, 0.848] (telephones), c. [5.631, 6.369].

Step-by-step solution

- Identify the problem

The task is to construct confidence intervals to estimate the population mean for different variables provided in the sample data: number of people per household, number of television sets and telephones per household, and hours of television viewing per day, all with a 90% confidence level.

- Find critical value for 90% confidence level

Use a Z-table or a statistical calculator to find the critical value for a 90% confidence interval. The critical value (Z) for a 90% confidence interval is approximately 1.645.

- Calculate margin of error for number of people per household

Compute the margin of error using the formula \(\text{Margin of Error} = Z \times \frac{s}{\sqrt{n}}\), where Z is the critical value, s is the standard deviation, and n is the sample size. For this problem: \(\text{Margin of Error} = 1.645 \times \frac{0.35}{\sqrt{178}} = 0.043\)

- Calculate confidence interval for number of people per household

Use the margin of error to calculate the confidence interval. The confidence interval is given by \(\text{Mean} \pm \text{Margin of Error}\). For this problem: Confidence Interval = 2.3 \pm 0.043, which is \([2.257, 2.343]\).

- Calculate margin of error for number of television sets per household

For the number of television sets per household: \(\text{Margin of Error} = 1.645 \times \frac{0.10}{\text{\textbackslash sqrt{178}}} = 0.012\)

- Calculate confidence interval for number of television sets per household

Confidence Interval for television sets: \(\text{Mean} \pm \text{Margin of Error} = 2.1 \pm 0.012\). This gives \([2.088, 2.112]\).

- Calculate margin of error for number of telephones per household

For the number of telephones per household: \(\text{Margin of Error} = 1.645 \times \frac{0.55}{\text{\textbackslash sqrt{178}}} = 0.068\)

- Calculate confidence interval for number of telephones per household

Confidence Interval for telephones: \(\text{Mean} \pm \text{Margin of Error} = 0.78 \pm 0.068\). This gives \([0.712, 0.848]\).

- Calculate margin of error for hours of television viewing per day

For the hours of television viewing per day: \(\text{Margin of Error} = 1.645 \times \frac{3.0}{\text{\textbackslash sqrt{178}}} = 0.369\)

- Calculate confidence interval for hours of television viewing per day

Confidence Interval for television viewing: \(\text{Mean} \pm \text{Margin of Error} = 6.0 \pm 0.369\). This gives \([5.631, 6.369]\).

Key concepts

population mean estimation

Population mean estimation is a fundamental concept in statistics. It involves using data from a sample to make inferences about the mean (average) of a larger population. In our exercise, the sample includes information from 178 households about various variables, like the number of people per household and hours of TV viewing. The goal is to make an educated guess about these averages for the entire population based on our sample data.

90% confidence level

A 90% confidence level indicates that we are 90% confident that the true population mean lies within our calculated interval. In simpler terms, if we were to take 100 different samples and build a confidence interval from each, approximately 90 of those intervals would contain the true population mean. This confidence level is slightly more conservative than the commonly used 95% level, giving a narrower range but still providing reasonable assurance. The critical value for a 90% confidence level, which is used in our calculations, is 1.645.

margin of error calculation

The margin of error quantifies the amount of random sampling error in our estimates. It's calculated using the formula: the formula \(\text{Margin of Error} = Z \times \frac{s}{\text{\text{\backslash}}sqrt{n}}\), where 'Z' is the critical value, 's' is the standard deviation, and 'n' is the sample size. The Z-value depends on our chosen confidence level. For a 90% confidence level, it's 1.645. The margin of error helps us create the range that forms our confidence interval. For instance, in our exercise, we calculated a margin of error of 0.043 for the number of people per household and 0.369 for daily TV viewing hours.

Z-table usage

The Z-table is a crucial tool in statistics for finding the critical values needed in confidence interval calculations. It shows the area (probability) under the standard normal curve corresponding to different Z-values. For a 90% confidence interval, the critical value 'Z' is 1.645. This value corresponds to a cumulative probability of 0.95 (covering the middle 90% of the distribution). Using a Z-table, or a statistical calculator, we can quickly lookup these critical values to use in our margin of error calculations. Proper Z-table usage ensures that we accurately build our confidence intervals, providing valid and reliable estimates of the population mean.