/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 Going back to school can be an e... [FREE SOLUTION] | 91Ó°ÊÓ

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Going back to school can be an expensive time for parents - second only to the Christmas holiday season in terms of spending (San Luis Obispo Tribune, August 18,2005\() .\) Parents spend an average of \(\$ 444\) on their children at the beginning of the school year stocking up on clothes, notebooks, and even iPods. Of course, not every parent spends the same amount of money and there is some variation. Do you think a data set consisting of the amount spent at the beginning of the school year for each student at a particular elementary school would have a large or a small standard deviation? Explain.

Short Answer

Expert verified
In this case, there would likely be a large standard deviation. The spending amount on the school supplies by different parents likely varies widely, leading to a wide spread around the average of \$444.

Step by step solution

01

Understanding Standard Deviation

The standard deviation is a measure of how spread out the values in a data set are around the mean (average), in this case, the average is \$444. If the amounts spent by all parents were similar (closer to \$444), then the standard deviation would be small. However, if the amounts spent by parents vary widely (some far less or far more than \$444), then the standard deviation would be large.
02

Looking at the given situation

As we do not have specific values, we must base our answer on the given situation where the spending at the beginning of the school year for students may vary largely since some parents might just fulfill the basic needs whereas others might go as far as buying iPods, etc. Hence there is a significant spread in the data.
03

Drawing a conclusion

Given this, we can infer that there would be a large standard deviation. The significant spread in the amount of money spent indicates that the spending varies widely around the average (\$444) among the parents.

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Key Concepts

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

Data Variation
When we talk about data variation, we're referring to how much the individual numbers in a data set differ from one another. In terms of school spending, this means looking at the differences in how much each parent spends on back-to-school items. Variability in data is important because it helps us understand how consistent or inconsistent the data points are. If there's a lot of variation, it suggests that parents are spending vastly different amounts, which might be due to their personal preferences and financial situations.

Data variation is usually measured using standard deviation. This tells us how much the amounts parents spend differ from the average spending. A low variation would mean most parents spend close to the same amount. On the other hand, high variation indicates that some parents spend significantly less or more than others.
Average Spending
The concept of average spending revolves around finding out what the common spending amount is among a group—in this case, parents buying back-to-school supplies. To determine this, you would sum up all the individual spending amounts and then divide by the number of people. This gives us the average, or mean, which is often used for budgeting and planning purposes.

For this exercise, an average spending of $444 was mentioned for parents during the back-to-school season. This figure helps us understand what a typical dollar amount spent looks like and provides a benchmark to gauge whether individual spending is above or below this line. While the average offers insight, it doesn't capture how varied the spending numbers might be, which is why standard deviation is also crucial.
Elementary School Data
Analyzing data from an elementary school provides a specific context for understanding spending patterns. In an elementary school, students might require specific items, such as notebooks, pencils, uniforms, and accessories, which can differ greatly in price. Each family approaches school shopping with unique needs and resources, resulting in different spending levels.

Elementary school data showcases real-world examples of how varied factors like income, family size, and personal priorities can influence spending. It's critical to collect and analyze this data to comprehend the wider socio-economic picture of the school group, which can then inform decisions on school programs and support systems.
Spending Analysis
Conducting a spending analysis involves looking at the spending behaviors of parents on school-related items. This requires gathering data, calculating averages, and measuring variations to understand spending habits better. Through this analysis, one can identify trends, such as whether parents are purchasing more tech gadgets like iPods over traditional supplies, which could indicate shifting priorities in education needs.

Spending analysis helps stakeholders—such as school administrators and parents themselves—make more informed decisions. By understanding how much families are spending and what they are spending on, schools can tailor their communication and assistance programs. It also aids in resource allocation, ensuring that students' essential needs are met without unnecessary financial strain on families. Overall, this analysis sheds light on the economic and social factors affecting school communities.

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Most popular questions from this chapter

Suppose that your younger sister is applying for entrance to college and has taken the SATs. She scored at the 83 rd percentile on the verbal section of the test and at the 94 th percentile on the math section of the test. Because you have been studying statistics, she asks you for an interpretation of these values. What would you tell her?

The chapter introduction gave the accompanying data on the percentage of those eligible for a lowincome subsidy who had signed up for a Medicare drug plan in each of 49 states (information was not available for Vermont) and the District of Columbia (USA Today, May 9,2006 ). $$ \begin{array}{llllllll} 24 & 27 & 12 & 38 & 21 & 26 & 23 & 33 \\ 19 & 19 & 26 & 28 & 16 & 21 & 28 & 20 \\ 21 & 41 & 22 & 16 & 29 & 26 & 22 & 16 \\ 27 & 22 & 19 & 22 & 22 & 22 & 30 & 20 \\ 21 & 34 & 26 & 20 & 25 & 19 & 17 & 21 \\ 27 & 19 & 27 & 34 & 20 & 30 & 20 & 21 \\ 14 & 18 & & & & & & \end{array} $$ a. Compute the mean for this data set. b. The article stated that nationwide, \(24 \%\) of those eligible had signed up. Explain why the mean of this data set from Part (a) is not equal to 24 . (No information was available for Vermont, but that is not the reason that the mean differs- the \(24 \%\) was calculated excluding Vermont.)

The following data are cost (in cents) per ounce for nine different brands of sliced Swiss cheese (www .consumerreports.org): \(\begin{array}{llllllll}29 & 62 & 37 & 41 & 70 & 82 & 47 & 52\end{array}\) a. Compute the variance and standard deviation for this data set. \(s^{2}=279.111 ; s=16.707\) b. If a very expensive cheese with a cost per slice of 150 cents was added to the data set, how would the values of the mean and standard deviation change?

The standard deviation alone does not measure relative variation. For example, a standard deviation of \(\$ 1\) would be considered large if it is describing the variability from store to store in the price of an ice cube tray. On the other hand, a standard deviation of \(\$ 1\) would be considered small if it is describing store-to-store variability in the price of a particular brand of freezer. A quantity designed to give a relative measure of variability is the coefficient of variation. Denoted by CV, the coefficient of variation expresses the standard deviation as a percentage of the mean. It is defined by the \(C V=100\left(\frac{s}{\bar{x}}\right)\) formula Consider two samples. Sample 1 gives the actual weight (in ounces) of the contents of cans of pet food labeled as having a net weight of 8 ounces. Sample 2 gives the actual weight (in pounds) of the contents of bags of dry pet food labeled as having a net weight of 50 pounds. The weights for the two samples are \(\begin{array}{lrrrrr}\text { Sample 1 } & 8.3 & 7.1 & 7.6 & 8.1 & 7.6 \\ & 8.3 & 8.2 & 7.7 & 7.7 & 7.5 \\ \text { Sample 2 } & 52.3 & 50.6 & 52.1 & 48.4 & 48.8 \\ & 47.0 & 50.4 & 50.3 & 48.7 & 48.2\end{array}\) a. For each of the given samples, calculate the mean and the standard deviation. b. Compute the coefficient of variation for each sample. Do the results surprise you? Why or why not?

The article "Taxable Wealth and Alcoholic Beverage Consumption in the United States" (Psychological Reports [1994]: \(813-814\) ) reported that the mean annual adult consumption of wine was 3.15 gallons and that the standard deviation was 6.09 gallons. Would you use the Empirical Rule to approximate the proportion of adults who consume more than 9.24 gallons (i.e., the proportion of adults whose consumption value exceeds the mean by more than 1 standard deviation)? Explain your reasoning.

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