91Ó°ÊÓ

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 (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 given data to estimate \(\pi\), the proportion of all houses that used at least 100 therms. d. Give a point estimate of the population median usage based on the given sample. Which statistic did you use?

Step by step solution

01

Compute a point estimate of \(\mu_{J}\)

Begin by calculating the mean (average) of the given data set. Sum up all the observations and divide this by the number of observations. To obtain the point estimate for \(\mu_{J}\), it is simply equal to the calculated mean.

02

Estimate \(\tau\)

Based on the assumption in the exercise, the population is the set of all houses (10,000) in the area. To estimate \(\tau\), the total amount of gas used by all these houses, multiply the calculated mean by the population size.

03

Estimate \(\pi\)

To estimate \(\pi\), the proportion of all houses that used at least 100 therms, count the number of houses that used 100 or more therms in the sample. Divide this count by the total number of houses in the sample.

04

Compute point estimate of the median usage

The median is the middle value when data points are arranged in ascending order. For this data set, there are 10 points. Therefore the median is the average of the 5th and 6th observations in the ordered list.

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Key Concepts

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

Mean Estimation

Estimating the mean is one of the fundamental tasks in statistics. It gives us a point estimate of the average value in a data set. To find the mean of a sample, you need to sum up all the observed values and then divide the total by the number of observations. For example, with our data on gas usage:

• Add up all the therm values: 103 + 156 + 118 + 89 + 125 + 147 + 122 + 109 + 138 + 99.
• The sum is 1206.
• Now divide this total by the number of houses, which is 10.

This gives us a mean of 120.6 therms. This mean acts as a point estimate for \(\mu_{J}\), the average gas usage for all houses in the area. It's easy to see that the mean provides a useful summary of the data, bridging the observed sample to the larger population.

Proportion Calculation

Calculating a proportion helps us understand the relative size of a subset within a larger group. In the exercise, we are interested in the proportion of houses that used at least 100 therms of gas. To calculate this, we:

• Count how many houses used 100 or more therms. In our data, the counts are 9 (103, 156, 118, 125, 147, 122, 109, 138, 99).
• There are 10 total observations.
• Divide the count of qualifying houses (9) by the total number of houses (10).

Thus, the proportion \(\pi\) is 0.9 or 90%. Proportion calculation is particularly useful in scenarios where we want to make predictions or infer trends about a population, based on sample data. It helps in distinguishing how frequently a particular trait or characteristic appears within the data set.

Median Estimation

The median gives a robust measure of central tendency and provides valuable insight, especially for skewed distributions. To estimate the median, arrange your data in ascending order:

• Ordered values: 89, 99, 103, 109, 118, 122, 125, 138, 147, 156.
• For ten pieces of data, the median will be the average of the 5th and 6th values.
• These values are 118 and 122.
• Average them to get the median: (118 + 122)/2 = 120.

This value, 120, is the point estimate of the population median. Median is less affected by extreme values (outliers) and thereby can reveal a more representative center of the dataset than the mean in certain conditions.

Sample Data Analysis

Analyzing a sample involves extracting key statistics to make informed estimates about a larger population. In our example, we examine a sample of gas usage data to predict broader trends and figures. Here's what we did:

• Calculated the mean to represent the average gas usage.
• Estimated the total gas usage for the entire population by scaling the sample mean.
• Determined the proportion of houses exceeding a specific threshold.
• Appraised the median to understand the central usage figure, unaffected by outliers.

Effective sample data analysis enables us to make educated guesses about the population from which the sample is drawn, improving our decision-making and strategic planning abilities. By leveraging mean, proportion, and median calculations, we succinctly interpret raw data and provide a snapshot into potential real-world scenarios.