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Chapter 14: Problem 1

As part of a sixth-grade class project the teacher brings to class a large jar containing 200 gumballs of two different colors: red and green. Andy is asked to draw a sample of his own choosing and estimate the number of red gumballs in the jar. Andy draws a sample of 25 gumballs, of which 8 are red and 17 are green. Use Andy's sample to estimate the number of red gumballs in the jar.

Short Answer

Expert verified
There are approximately 64 red gumballs in the jar.

Step by step solution

01

Determine the Proportion of Red Gumballs in the Sample

Calculate the proportion of red gumballs in Andy's sample by dividing the number of red gumballs by the total number of gumballs in the sample. Let's let the number of red gumballs be denoted by R and total gumballs in the sample by S.
02

Calculate the Estimated Total Number of Red Gumballs

Assuming the sample proportion is representative of the entire jar, multiply the proportion of red gumballs in the sample by the total number of gumballs in the jar to estimate the total number of red gumballs in the jar. Let the total number of gumballs in the jar be denoted by T.
03

Write the Estimated Number of Red Gumballs

The result from Step 2 is the estimate of the total number of red gumballs in the jar.

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Key Concepts

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

Proportion Calculation
When it comes to statistics, calculating proportions is an essential skill. For example, in Andy's class project involving a jar of gumballs, the proportion calculation starts by identifying the relative frequency of the red gumballs within his sample.

This is done by dividing the number of red gumballs, which we will denote as \(R\), by the total number of gumballs in Andy's sample, \(S\). Written in mathematical terms, the proportion of red gumballs is represented by the formula: \(\frac{R}{S}\). By plugging in the values from Andy's selection 鈥� 8 red gumballs out of a sample of 25 鈥� the calculation is a straight-forward \(\frac{8}{25}\). This result is a key piece in estimating the population proportion and understanding the likelihood of picking a red gumball from the jar randomly.
Sample Representation
Now, let's unpack the concept of sample representation. In the context of the class project, Andy's selection is a small, manageable representation of the entire jar's contents. It's critical when using samples to make assumptions about larger populations that the sample is as representative as possible.

In an ideal world, each color gumball would have an equal chance of being selected, making the sample random. This process minimizes bias and makes our estimation more reliable. Andy's sample of 25 gumballs needs to reflect the larger population of 200 gumballs accurately for the proportion calculation to correctly predict the total number of red gumballs. A sample that is not representative could significantly sway the results and lead to inaccurate conclusions about the entire population.
Estimation Techniques
The last puzzle piece is understanding estimation techniques. After calculating the sample proportion, as Andy did with the red gumballs, this proportion is then used to estimate the proportion in the total population. The essence of the technique is to scale up the findings from the sample to the entire population.

An estimator will consider the sample proportion as a miniature model of the population. If the sample is free from bias, it gives us a mini-version of what the entire jar looks like. Andy then multiplies his sample proportion by the total number of gumballs: \(\frac{R}{S} \times T\), where \(T\) is the total number of gumballs in the jar. Through this estimation technique, we get an approximate total count of red gumballs in the jar, giving us valuable insight into the population without counting every single gumball.

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Most popular questions from this chapter

Madison County has a population of 34,522 people. The county hospital is interested in estimating the number of people in the county with blood-type \(A-\) To do this they test blood samples from 253 patients. Out of this group, 17 have blood-type \(A-\) Use this sample to estimate the number of people in Madison County with blood-type A-.

Refer to the following story: The dean of students at Tasmania State University wants to determine how many undergraduates at TSU are familiar with a new financial aid program offered by the university. There are 15,000 undergraduates at TSU, so it is too expensive to conduct a census. The following sampling method is used to choose a representative sample of undergraduates to poll. Start with the registrar's alphabetical listing containing the names of all undergraduates. Randomly pick a number between \(l\) and \(100,\) and count that far down the list. Take that name and every I00th name after it. For example, if the random number chosen is \(73,\) then pick the \(73 \mathrm{rd}, 173 \mathrm{rd}, 273 \mathrm{rd}\) and so forth, names on the list. (The sampling method illustrated in this survey is known as systematic sampling.) (a) Explain why the method used for choosing the sample is not simple random sampling. (b) If \(100 \%\) of those responding claimed that they were not familiar with the new financial aid program offered by the university, is this result more likely due to sampling variability or to sample bias? Explain.

Refer to the following story: The city of Cleansburg has 8325 registered voters. There is an election for mayor of Cleansburg, and there are three candidates for the position: Smith, Jones, and Brown. The day before the election a tele. phone poll of 680 randomly chosen registered voters produced the following results: 306 people surveyed indicated that they would vote for Smith, 272 indicated that they would vote for Jones and I02 indicated that they would vote for Brown. Given that in the actual election Smith received \(42 \%\) of the vote. Jones \(43 \%\) of the vote, and Brown \(15 \%\) of the vote, find the sampling errors in the survey expressed as percentages.

Today, most consumer marketing surveys are conducted by telephone. In selecting a sample of households that are representative of all the households in a given geographical area, the two basic techniques used are (1) randomly selecting telephone numbers to call from the local telephone directory or directories and (2) using a computer to randomly generate seven-digit numbers to try that are compatible with the local phone numbers. (a) Briefly discuss the advantages and disadvantages of each technique. In your opinion, which of the two will produce the more reliable data? Explain. (b) Suppose that you are trying to market burglar alarms in New York City. Which of the two techniques for selecting the sample would you use? Explain your reasons.

Refer to the following story: An orange grow. er wishes to compute the average yield from his orchard. The orchard contains three varieties of trees: \(50 \%\) of his trees are of variety \(A, 25 \%\) of variety \(B,\) and \(25 \%\) of variety \(C\) (a) Suppose that in his survey, the grower found that each tree of variety A averages 100 oranges, each tree of variety \(\mathrm{B}\) averages 50 oranges, and each tree of varietyC averages 70 oranges. Estimate the average yield per tree of his orchard. (b) Is the yield you found in (a) a parameter or a statistic? Explain.
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