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Chapter 8: Problem 4

Consider the following statement: A sample of size 100 was selected from those admitted to a particular college in fall 2017 . The proportion of these 100 who were transfer students is \(\mathbf{0} . \mathbf{3 8}\). a. Is the number that appears in boldface in this statement a sample proportion or a population proportion? b. Which of the following use of notation is correct, \(p=0.38\) or \(\hat{p}=0.38 ?\)

Short Answer

Expert verified
a. The boldface number 0.38 represents a sample proportion. b. The correct notation is \(\hat{p}=0.38\).

Step by step solution

01

Interpret the Statement

In the statement, we have a sample of size 100 from those admitted to the college in fall 2017. From these 100 students, 38% (which is 0.38 in decimal form) were transfer students.
02

Determine the Type of Proportion

Since the information is taken from a sample of students (sample size 100), the proportion 0.38 represents a sample proportion. a. The boldface number 0.38 represents a sample proportion.
03

Choose the Correct Notation

The correct notation for a sample proportion is \(\hat{p}\). So the appropriate notation is: b. \(\hat{p}=0.38\)

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Key Concepts

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

Understanding Statistics
Statistics plays a crucial role in gathering, analyzing, and interpreting data in various fields. It enables us to make sense of numerical data and extract meaningful patterns. In educational settings, statistics empower students to understand real-world problems through the lens of data. For instance, when analyzing the characteristics of a college's incoming class, statistics can guide us on how to collect data, evaluate it, and make informed decisions.

Statistical analysis often commences with a question or hypothesis, and then researchers collect data to address it. This data can come in the form of a sample, which is a subset of the larger population we are interested in understanding. For instance, gathering information on 100 students from a college to determine the proportion that are transfer students. This sample provides concrete numbers that can be analyzed statistically to estimate characteristics of the overall student population, providing educators and administrators with insights that can shape policies and support services.
Population Proportion

Defining Population Proportion

Population proportion represents the fraction or percentage of members in a whole population that possess a particular characteristic. It is symbolically denoted as 'p'. When educational researchers want to understand the characteristics of an entire college's enrollment, they would use population proportion to indicate the percentage of, for example, transfer students in all admissions.

However, it is often not feasible or practical to collect data from every single individual in the population. This is where a sample proportion ((hat{p})) becomes valuable, as it estimates the population proportion based on data from a sample. While it is not the true population proportion, it gives a useful approximation that can assist educational stakeholders in making decisions without needing to survey every student.
Statistical Notation

Sample Proportion Notation

Statistical notation is the standardized language used by statisticians to succinctly document statistical calculations and concepts. In the context of sample proportions, the correct notation is crucial. The sample proportion is expressed as (hat{p}), where the 'hat' symbol above 'p' distinguishes it from the population proportion, 'p'.

Using the correct notation helps avoid confusion and ensures clarity in communication amongst statisticians and those interpreting the data. For example, when we state that the sample proportion of transfer students in a college is (hat{p}=0.38), it clearly conveys that this figure is derived from a sample, and not the entire student population. This distinction is essential, especially when conveying findings to stakeholders who might not have a statistical background, but who need to understand the implications of data analysis in policy making or strategic planning.

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Most popular questions from this chapter

A random sample of 1000 students at a large college included 428 who had one or more credit cards. For this sample, \(\hat{p}=\frac{428}{1000}=0.428 .\) If another random sample of 1000 students from this university were selected, would you expect that \(\hat{p}\) for that sample would also be 0.428 ? Explain why or why not.

Suppose that \(20 \%\) of the customers of a cable television company watch the Shopping Channel at least once a week. The cable company does not know the actual proportion of all customers who watch the Shopping Channel at least once a week and is trying to decide whether to replace this channel with a new local station. The company plans to take a random sample of 100 customers and to use \(\hat{p}\) as an estimate of the population proportion. a. Show that \(\sigma_{p},\) the standard deviation of \(\hat{p},\) is equal to 0.040 b. If for a different sample size, \(\sigma_{p}=0.023,\) would you expect more or less sample-to-sample variability in the sample proportions than when \(n=100 ?\) c. Is the sample size that resulted in \(\sigma_{\hat{p}}=0.023\) larger than 100 or smaller than \(100 ?\) Explain your reasoning.

Suppose that a particular candidate for public office is favored by \(48 \%\) of all registered voters in the district. A polling organization will take a random sample of 500 of these voters and will use \(\hat{p},\) the sample proportion, to estimate \(p\). a. Show that \(\sigma_{\hat{p}}\), the standard deviation of \(\hat{p},\) is equal to 0.022 . b. If for a different sample size, \(\sigma_{j}=0.071,\) would you expect more or less sample-to-sample variability in the sample proportions than when \(n=500 ?\) c. Is the sample size that resulted in \(\sigma_{\hat{p}}=0.071\) larger than 500 or smaller than \(500 ?\) Explain your reasoning.

A random sample is to be selected from a population that has a proportion of successes \(p=0.25 .\) Determine the mean and standard deviation of the sampling distribution of \(\hat{p}\) for each of the following sample sizes: a. \(n=10\) d. \(n=50\) b. \(n=20\) e. \(n=100\) c. \(n=30\) f. \(n=200\)

A random sample of size 300 is to be selected from a population. Determine the mean and standard deviation of the sampling distribution of \(\hat{p}\) for each of the following population proportions. a. \(p=0.20\) b. \(p=0.45\) c. \(p=0.70\) d. \(p=0.90\)
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