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Chapter 3: Problem 7

Find the population mean or sample mean as indicated. Sample: 20,13,4,8,10

Short Answer

Expert verified
The sample mean is 11.

Step by step solution

01

Understand the Problem

We are given a sample set of data and need to find the sample mean. The given sample is: 20, 13, 4, 8, 10.
02

Sum the Data Points

Add up all the data points in the sample. \[ 20 + 13 + 4 + 8 + 10 = 55 \]
03

Count the Number of Data Points

Count how many data points are in the sample. There are 5 data points.
04

Calculate the Sample Mean

Divide the sum of the data points by the number of data points. \[ \text{Sample Mean} = \frac{\text{Sum of Data Points}}{\text{Number of Data Points}} = \frac{55}{5} = 11 \]

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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 measure of the average value of a set of data points. It is calculated by summing all the data points in the sample and then dividing that sum by the number of data points. This is a fundamental concept in statistics and is especially useful in making inferences about a population based on a sample. For instance, if we have a sample data set: 20, 13, 4, 8, and 10, we can find the sample mean to understand the central tendency of this data. Calculating the sample mean helps to summarize the data with a single value that represents the 'central' or 'average' value of the dataset.
Data Points
Data points are individual values in a dataset. When we talk about calculating the sample mean, we're dealing with each value in the sample. In our example, the data points are 20, 13, 4, 8, and 10. It's important to understand that each data point contributes equally to the sum, regardless of its magnitude. When counting data points, it helps to ensure that we don't miss any values, as the number of data points (N) is crucial for calculating the sample mean accurately. In our case, there are 5 data points.
Sum of Data Points
Summing the data points means adding all the individual values in a dataset together. This is a simple yet essential step in calculating the sample mean. For example, given the data points 20, 13, 4, 8, and 10, you add them together: 20 + 13 + 4 + 8 + 10 = 55 Once you have the total sum of the data points, you can proceed to determine the mean by dividing this sum by the number of data points. The sum of the data points gives a crucial piece of information about the dataset and acts as the intermediary step in most basic statistical calculations, including variance and standard deviation.

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Most popular questions from this chapter

Morningstar is a mutual fund rating agency. It ranks a fund's performance by using one to five stars. A one-star mutual fund is in the bottom \(10 \%\) of its investment class; a five-star mutual fund is at the 90 th percentile of its investment class. Interpret the meaning of a five-star mutual fund.

The highest batting average ever recorded in Major League Baseball was by Ted Williams in 1941 when he hit \(0.406 .\) That year, the mean and standard deviation for batting average were 0.2806 and \(0.0328 .\) In 2014 Jose Altuve was the American League batting champion, with a batting average of \(0.341 .\) In \(2014,\) the mean and standard deviation for batting average were 0.2679 and \(0.0282 .\) Who had the better year relative to his peers, Williams or Altuve? Why?

Explain how to determine the shape of a distribution using the box plot and quartiles.

Lawrence Summers (former Secretary of the Treasury and former president of Harvard) infamously claimed that women have a lower standard deviation IQ than men. He went on to suggest that this was a potential explanation as to why there are fewer women in top math and science positions. Suppose an IQ of 145 or higher is required to be a researcher at a top-notch research institution. Use the idea of standard deviation, the Empirical Rule, and the fact that the mean and standard deviation IQ of humans is 100 and 15 , respectively, to explain Summers' argument.

It is well documented that active maternal smoking during pregnancy is associated with lower-birth-weight babies. Researchers wanted to determine if there is a relationship between paternal smoking habits and birth weight. The researchers administered a questionnaire to each parent of newborn infants. One question asked whether the individual smoked regularly. Because the survey was administered within 15 days of birth, it was assumed that any regular smokers were also regular smokers during pregnancy. Birth weights for the babies (in grams) of nonsmoking mothers were obtained and divided into two groups, nonsmoking fathers and smoking fathers. The given data are representative of the data collected by the researchers. The researchers concluded that the birth weight of babies whose father smoked was less than the birth weight of babies whose father did not smoke. $$ \begin{array}{lll|lll} &{\text { Nonsmokers }} & &&{\text { Smokers }} \\ \hline 4194 & 3522 & 3454 & 3998 & 3455 & 3066 \\ \hline 3062 & 3771 & 3783 & 3150 & 2986 & 2918 \\ \hline 3544 & 3746 & 4019 & 4216 & 3502 & 3457 \\ \hline 4054 & 3518 & 3884 & 3493 & 3255 & 3234 \\ \hline 4248 & 3719 & 3668 & 2860 & 3282 & 2746 \\ \hline 3128 & 3290 & 3423 & 3686 & 2851 & 3145 \\ \hline 3471 & 4354 & 3544 & 3807 & 3548 & 4104 \\ \hline 3994 & 2976 & 4067 & 3963 & 3892 & 2768 \\ \hline 3732 & 3823 & 3302 & 3769 & 3509 & 3629 \\ \hline 3436 & 3976 & 3263 & 4131 & 3129 & 4263 \\ \hline \end{array} $$ (a) Is this an observational study or a designed experiment? Why? (b) What is the explanatory variable? What is the response variable? (c) Can you think of any lurking variables that may affect the results of the study? (d) In the article, the researchers stated that "birthweights were adjusted for possible confounders \(\ldots .\) "What does this mean? (e) Determine summary statistics (mean, median, standard deviation, quartiles) for each group. (f) Interpret the first quartile for both the nonsmoker and smoker group. (g) Draw a side-by-side box plot of the data. Does the side-byside boxplot confirm the conclusions of the study?
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