91Ó°ÊÓ

A random sample of 10 houses heated with natural gas in a particular area, is selected, and the amount of gas (in therms) used during the month of January is determined for each house. The resulting observations are as follows: \(\begin{array}{llllllllll}103 & 156 & 118 & 89 & 125 & 147 & 122 & 109 & 138 & 99\end{array}\) a. Let \(\mu_{J}\) denote the average gas usage during January by all houses in this area. Compute a point estimate of \(\mu_{J}\) b. Suppose that 10,000 houses in this area use natural gas for heating. Let \(\tau\) denote the total amount of gas used by all of these houses during January. Estimate \(\tau\) using the given data. What statistic did you use in computing your estimate? c. Use the data in Part (a) to estimate \(p\), the proportion of all houses that used at least 100 therms. d. Give a point estimate of the population median us- age based on the sample of Part (a). Which statistic did you use?

Step by step solution

01

Calculate the point estimate of \(\mu_{J}\)

The point estimate of \(\mu_{J}\), the average usage of gas during January, is simply the arithmetic mean of the provided data set. That is, sum up all the 10 observations and divide by 10 to get the average.

02

Estimate \(\tau\)

To estimate \(\tau\), the total amount of gas used by all houses, multiply the point estimate of \(\mu_{J}\) obtained in Step 1 by the total number of houses, which is 10,000 in this case. This is because \(\mu_{J}\) is considered representative of the average consumption per house in the locality.

03

Estimate the proportion \(p\)

To estimate \(p\), the proportion of houses that used at least 100 therms, count the number of observations greater than or equal to 100 and divide by the total number of observations, i.e. 10.

04

Estimate the population median usage

To estimate the population median usage, sort the data set in ascending order, and since it contains 10 observations which is an even number, the median is a mean of the two central numbers. That is, take the average of the 5th and the 6th observation.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Sample Mean

Understanding the sample mean is fundamental when analyzing data. The sample mean, often represented by \( \bar{x} \), is a very common point estimate used to infer the average of a population. In the context of a real-world scenario - for example, the average gas usage by houses during January - we use the observations from our sample to calculate it. To find the sample mean, you sum up all the observed values and divide by the number of observations.

For the exercise provided, you would add the gas usage of all 10 houses together and divide this total by 10. It's a straightforward approach, but it plays a crucial role in statistics as it sets the stage for more complex analyses like regressions and hypothesis testing. It's important to remember, though, the sample mean is only an estimate of the true population mean, and it’s accuracy depends on how representative the sample is of the entire population.

Population Total Estimation

When we talk about population total estimation, we're venturing beyond our sample to make predictions about the total quantity of interest within an entire population. This typically involves multiplying the sample mean by the population size.

In our exercise, after calculating the average gas usage from our sample, we estimate the total gas usage, \( \tau \), for all 10,000 houses by taking our sample mean as a representation of the average usage per house. This extrapolates the data from the 10 observed houses to all the houses using gas in the area. It's an efficient method for estimation, but we must be cautious and consider the variability and potential bias within our sample. If our sample is not representative, our estimate might be less accurate.

Proportion Estimate

Estimating proportions is another key statistical task that helps us understand the characteristics of our population. A proportion estimate is simply the fraction of the sample that meets a certain criterion.

In this exercise, we're asked to estimate \( p \), the proportion of houses that used at least 100 therms of gas. To do this, we count how many houses in our sample used 100 or more therms, and divide by the total number of houses in the sample. Proportion estimates like this can provide a quick snapshot of a particular subset within your data, and are especially useful in opinion polls and quality control.

Population Median

The population median is a measure of central tendency that indicates the middle value of a dataset when ordered from smallest to largest. Estimating the population median from a sample involves sorting your sample data and identifying the middle value or, if the sample size is even, taking the average of the two middle values.

This was exactly the process used in the exercise, where after arranging the gas usage amounts in ascending order, we calculated the median by averaging the 5th and 6th observations. The median is less affected by outliers and skewed data compared to the mean, and thus it provides another perspective on the center of the data. It’s a valuable statistic when you're dealing with data that isn’t symmetrically distributed or when you're interested in the typical experience in a population, rather than the overall level.