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Chapter 2: Problem 6

A distribution of measurements is relatively mound-shaped with a mean of 50 and a standard deviation of \(10 .\) Use this information to find the proportion of measurements in the intervals given in Exercises \(6-11 .\) Between 40 and 60

Short Answer

Expert verified
Answer: Approximately 68%.

Step by step solution

01

Identify the mean and standard deviation

The given information tells us that the mean (\(\mu\)) is 50 and the standard deviation (\(\sigma\)) is 10.
02

Determine the range of one standard deviation from the mean

We are asked to find the proportion of measurements in the interval between 40 and 60. Since the mean is 50 and the standard deviation is 10, one standard deviation from the mean would be 50 卤 10, which results in a range of 40 to 60.
03

Apply the empirical rule

According to the empirical rule for mound-shaped distributions, approximately 68% of the data falls within one standard deviation from the mean. We have already determined that our given interval, 40 to 60, falls within one standard deviation from the mean.
04

State the proportion of measurements

Based on the empirical rule, the proportion of measurements in the interval between 40 and 60 is approximately 68%.

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Key Concepts

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

Standard Deviation
The concept of standard deviation is essential in statistics as it gives you a sense of how spread out a set of measurements is around the mean. Think of it like a measure of variability or dispersion; the higher the standard deviation, the more spread out the values in the data set. In a perfectly uniform distribution, the standard deviation would be zero because every value is exactly the same as the mean.
Mound-Shaped Distribution
When we talk about a mound-shaped distribution, we're picturing a graph that resembles a mound, or hill. The highest point corresponds to the mean, and the graph is symmetrical around this peak. As you move away from the mean, the frequency of the data points (how often they occur) decreases, producing the slopes of the mound. This shape indicates that most data points are clustered around the center, with fewer outliers at the extremes.
Normal Distribution
A mound-shaped distribution is often referred to as a normal distribution, especially when it meets some specific mathematical criteria. This bell-shaped curve is predictable and follows the empirical rule, which states that around 68% of data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and almost all (99.7%) fall within three standard deviations. It's a fundamental concept in statistics because many statistical tests are based on the assumption that the data is normally distributed.
Mean in Statistics
The mean, often called the average, is the central value in a data set and is calculated by adding all the measurements together and dividing by the number of measurements. In our context, the mean is like the balancing point of the mound-shaped distribution. It's important to understand that the mean is just one type of average, and in some cases, such as when outliers are present, other types like the median or mode might give you a better sense of the 'middle' of the data.
Statistical Intervals
Statistical intervals are ranges that capture a certain proportion of data in a distribution. For example, in the normal distribution, intervals defined by one, two, or three standard deviations from the mean can tell us the probability of finding a data point within those ranges. These intervals are incredibly useful when making predictions or inferences about populations based on sample data, as they give us a controlled way to discuss the variability and expected outcomes.

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Most popular questions from this chapter

To estimate the amount of lumber in a tract of timber, an owner randomly selected seventy 15 -by-15-meter squares, and counted the number of trees with diameters exceeding 1 meter in each square. The data are listed here: $$ \begin{array}{rrrrrrrrrr} 7 & 8 & 7 & 10 & 4 & 8 & 6 & 8 & 9 & 10 \\ 9 & 6 & 4 & 9 & 10 & 9 & 8 & 8 & 7 & 9 \\ 3 & 9 & 5 & 9 & 9 & 8 & 7 & 5 & 8 & 8 \\ 10 & 2 & 7 & 4 & 8 & 5 & 10 & 7 & 7 & 7 \\ 9 & 6 & 8 & 8 & 8 & 7 & 8 & 9 & 6 & 8 \\ 6 & 11 & 9 & 11 & 7 & 7 & 11 & 7 & 9 & 13 \\ 10 & 8 & 8 & 5 & 9 & 9 & 8 & 5 & 9 & 8 \end{array} $$ a. Construct a relative frequency histogram to describe the data. b. Calculate the sample mean \(\bar{x}\) as an estimate of \(\mu,\) the mean number of trees for all 15 -by-15-meter squares in the tract. c. Calculate \(s\) for the data. Construct the intervals \(\bar{x} \pm s, \bar{x} \pm 2 s,\) and \(\bar{x} \pm 3 s .\) Calculate the percentage of squares falling into each of the three intervals, and compare with the corresponding percentages given by the Empirical Rule and Tchebysheff's Theorem.

A distribution of measurements has a mean of 75 and a standard deviation of \(5 .\) You know nothing else about the size or shape of the data. Use this information to find the proportion of measurements in the intervals given. Between 62.5 and 87.5

Find the sample mean and the sample standard deviation and calculate the z-scores for the largest and smallest observations. Are there any unusually large or small observations? The weights (in pounds) of 27 packages of ground beef are listed here in order from smallest to largest. $$ \begin{array}{rrrrrrr} .75 & .83 & .87 & .89 & .89 & .89 & .92 \\ .93 & .96 & .96 & .97 & .98 & .99 & 1.06 \\ 1.08 & 1.08 & 1.12 & 1.12 & 1.14 & 1.14 & 1.17 \\ 1.18 & 1.18 & 1.24 & 1.28 & 1.38 & 1.41 & \end{array} $$

Mathematics achievement test scores for 400 students had a mean and a variance equal to 600 and \(4,900,\) respectively. If the distribution of test scores was mound-shaped, approximately how many scores would fall in the interval 530 to 670 ? Approximately how many scores would fall in the interval 460 to \(740 ?\)

For the data sets calculate the mean, the median, and the mode. Locate these measures on a dotplot. \(n=8\) measurements: 3,2,5,6,4,4,3,5
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