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Chapter 6: Problem 7

a. A random sample of 10 houses in a particular area, each of which is heated with natural gas, is selected and the amount of gas (therms) used during the month of January is determined for each house. The resulting observations are \(103,156,118,89,125,147,122,109,138,99\). Let \(\mu\) denote the average gas usage during January by all houses in this area. Compute a point estimate of \(\mu\). b. Suppose there are 10,000 houses in this area that 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 data of part (a). What estimator 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 usage (the middle value in the population of all houses) based on the sample of part (a). What estimator did you use?

Short Answer

Expert verified
a. 120.6 therms b. 1,206,000 therms, using sample mean c. 0.9 d. 120; sample median

Step by step solution

01

Calculate the Sample Mean

To compute a point estimate of the average gas usage, we calculate the sample mean. Add up all the usage values and divide by the number of observations. The calculation is as follows: \[ \text{Sample Mean} = \frac{103 + 156 + 118 + 89 + 125 + 147 + 122 + 109 + 138 + 99}{10} = \frac{1206}{10} = 120.6 \text{ therms} \]So, the point estimate of \( \mu \) is 120.6 therms.
02

Estimate Total Gas Usage

The total amount of gas used by all houses, \( \tau \), can be estimated by multiplying the average usage by the total number of houses. Using the point estimate from part (a), \[ \tau = 120.6 \times 10,000 = 1,206,000 \text{ therms} \]We used the estimator \( \hat{\tau} = n \times \bar{x} \), where \( n \) is the total number of houses and \( \bar{x} \) is the sample mean.
03

Estimate Proportion of Usage

To estimate the proportion \( p \) of houses using at least 100 therms, count the number of sample houses with usage \( \geq 100 \) and divide by the total number of observations. The calculation is as follows:Number of houses with usage \( \geq 100 \): 9 Proportion, \( p \): \[ p = \frac{9}{10} = 0.9 \]So, the estimated proportion is 0.9.
04

Estimate the Median Usage

The median is the middle value of an ordered dataset. First, arrange the usage values in increasing order: \[ 89, 99, 103, 109, 118, 122, 125, 138, 147, 156 \]Since there are 10 values, the median is the average of the 5th and 6th values: \[ \text{Median} = \frac{118 + 122}{2} = 120 \]We used the sample median as the estimator for the population median.

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Key Concepts

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

Sample Mean
The sample mean is a foundational concept in statistics used to estimate the average value of a dataset. To compute the sample mean, you sum up all the individual values and then divide by the number of observations.
In the context of the exercise, we calculated the sample mean for gas usage as follows:
  • Add all the gas usage values: \(103 + 156 + 118 + 89 + 125 + 147 + 122 + 109 + 138 + 99 = 1206\) therms.
  • Divide by the number of houses: \(\frac{1206}{10} = 120.6\) therms.
This gives us a sample mean of 120.6 therms, serving as a point estimate for \(\mu\), the average gas usage of all houses.The sample mean is crucial because it provides a simple yet effective way of summarizing data. It serves as an unbiased estimator of the population mean when samples are random and identically distributed.
Point Estimation
Point estimation is a technique in statistics where we use sample data to provide a single best estimate of a population parameter. In the exercise, the point estimates were derived for the average gas usage (the sample mean) and the total gas usage.Point estimates are extremely useful when we need quick assessments:
  • They provide a specific value rather than a range.
  • They are easy to compute and interpret.
The accuracy and reliability depend on sample size and selection method. In the exercise, the point estimate of the total gas usage \(\tau\) was computed by:\[\tau = 120.6 \times 10,000 = 1,206,000\text{ therms}\]Here, \(n = 10,000\), representing the total houses, and \(\bar{x}\), the sample mean. This technique allows statisticians and analysts to make informed decisions off of sample data quickly.
Population Median
The population median refers to the middle value in the entire dataset, dividing it into two equal halves. In our exercise, estimating the population median was based on the sample's median due to practical constraints on data collection.To find the median from the sample of 10 observations:
  • First, organize the data in ascending order: 89, 99, 103, 109, 118, 122, 125, 138, 147, 156.
  • Since there is an even number of data points, average the 5th and 6th values.
  • The calculation is: \(\frac{118 + 122}{2} = 120\).
Therefore, the sample median was used as an estimator of the population median. The median offers robustness to extreme values (outliers), providing more central tendency information than the mean in skewed distributions.
Proportion Estimation
Proportion estimation deals with estimating the fraction of a population that shares a particular attribute. Contrasting with the mean, which assesses average behavior, proportion estimation aims to understand prevalence.In the exercise, we determined the proportion of houses using at least 100 therms. Steps involved:
  • Count houses with usage \(\geq 100\): 9 out of 10.
  • Compute the proportion: \(\frac{9}{10} = 0.9\).
The calculated proportion of 0.9 indicates that 90% of the sampled houses used at least 100 therms.Proportion estimation is particularly useful in survey analysis and studies, helping with decisions in policy-making, marketing strategies, and more. It provides a way to measure outcomes when dealing with categorical data.

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Most popular questions from this chapter

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