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Chapter 16: Problem 8

Campus sample For her final project, Stacy plans on surveying a random sample of 50 students on whether they plan to go to Florida for spring break. From past years, she guesses that about \(10 \%\) of the class goes. Is it reasonablefor her to use a Normal model for the sampling distribution of the sample proportion? Why or why not?

Short Answer

Expert verified
No, it is not reasonable for Stacy to use a Normal model for the sampling distribution of the sample proportion as the success-failure condition is not met because the number of successes (\(np = 5\)) is less than 10.

Step by step solution

01

Check Binary Condition

The outcome is binary - The question is whether they plan to go to Florida for spring break or not, so each student either plans to go (success) or does not plan to go (failure).
02

Check Independence Condition

Given that the sample size is 50 and assuming that this is less than 10% of all students, it can be assumed that the sampling is done randomly and independently.
03

Check Success-Failure Condition

The estimated proportion of success (students planning to go to Florida for spring break) is 10%, that is, \(p = 0.10\). Hence, the estimated proportion of failures \(q = 1 - p = 0.90\). Multiply these percentages by the sample size (50) to confirm if they're 鈮� 10. So, \(np = 50 * 0.10 = 5\) and \(nq = 50 * 0.90 = 45\). The success condition does not hold (5 < 10) while the failure condition does hold (45 > 10).

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Key Concepts

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

Understanding the Binary Condition
When talking about sampling in statistics, the term binary condition refers to scenarios where there are only two possible outcomes. In the context of Stacy's survey about spring break plans, these two outcomes are either a student plans to go to Florida (which we can call a 'success') or does not plan to go (a 'failure').

This distinction is critical for determining the type of statistical model we can apply, as certain models, like the binomial or the Normal model, require a binary setup to make sense of the data. It's like flipping a coin - you only have heads or tails, no in-betweens. Stacy has correctly identified that her survey question meets this binary condition.
Independence Condition In Sampling
The independence condition is a cornerstone for reliable statistical calculations. It means each selection in the sample does not influence another, which is often satisfied if each individual is randomly selected and forms less than 10% of the population.

Imagine plucking apples from a tree - if plucking one doesn't affect which apple you pluck next, then the selections are independent. For Stacy's survey, we assume random sampling and that her 50 students are a fraction of the total student body, thus satisfying the independence condition. This ensures that results from one student do not bias another's response.
Delving into the Success-Failure Condition
The success-failure condition is pivotal to ascertain the validity of using a Normal model for binomial situations. For a sample size 'n' and the probability of success 'p', this condition requires both 'np' and 'n(1-p)' to be at least 10.

Understand it like this - if too few 'successes' or 'failures' are expected, the distribution of sample proportion will be skewed rather than Normal. Since Stacy's calculation gave her 5 expected successes, which is less than the minimum of 10, the success-failure condition fails. With this condition unmet, the use of Normal distribution is questionable.
Normal Model: When to Use It
A Normal model is a bell-shaped curve representing data distribution and it's very handy when dealing with probabilities and statistics. However, it hinges on certain assumptions like having a large enough number of successes and failures (as addressed in the success-failure condition) and the sample observations being independent.

Although popular for its simplicity and ease of use, particularly in predicting probabilities, the Normal model isn't the go-to for all scenarios. Like Stacy's survey, if expected successes (or failures) are too limited, the model might not accurately represent the data distribution, leading to incorrect conclusions.
Sample Proportion and Its Importance
In Stacy's context, sample proportion represents the part of her survey participants planning to go to Florida for spring break. Mathematically, it's determined by dividing the number of 'successes' in the sample by the total sample size.

It's a key piece of information as it provides an estimate of the actual proportion in the entire population - think of it like a snapshot from a bigger picture. A well-calculated sample proportion is vital for making predictions or inferences about the larger group, making it a fundamental concept in sample surveys and the field of inferential statistics.

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Most popular questions from this chapter

Send money When they send out their fundraising letters, a philanthropic organization typically gets a return from about \(5 \%\) of the people on their mailing list. To see what the response rate might be for future appeals, they did a simulation using samples of size \(20,50,100,\) and 200 . For each sample size, they simulated 1000 mailings with success rate \(p=0.05\) and constructed the histogram of the 1000 sample proportions, shown below. Explain what these histograms show about the sampling distribution model for sample proportions. Be sure to talk about shape, center, and spread.

Conditions For each situation described below, identify the population and the sample, explain what \(p\) and \(\hat{p}\) represent, and tell whether the methods of this chapter can be used to create a confidence interval. a. Police set up an auto checkpoint at which drivers are stopped and their cars inspected for safety problems. They find that 14 of the 134 cars stopped have at least one safety violation. They want to estimate the percentage of all cars that may be unsafe. b. A TV talk show asks viewers to register their opinions on prayer in schools by logging on to a website. Of the 602 people who voted, 488 favored prayer in schools. We want to estimate the level of support among the general public. c. A school is considering requiring students to wear uniforms. The PTA surveys parent opinion by sending a questionnaire home with all 1245 students; 380 surveys are returned, with 228 families in favor of the change. d. A college admits 1632 freshmen one year, and four years later, 1388 of them graduate on time. The college wants to estimate the percentage of all their freshman enrollees who graduate on time.

Website An investment company is planning to upgrade the mobile access to their website, but they'd like to know the proportion of their customers who access it from their smartphones. They draw a random sample of 200 from customers who recently logged in and check their IP address. Suppose that the true proportion of smartphone users is \(36 \%\) a. What would you expect the shape of the sampling distribution for the sample proportion to be? b. What would be the mean of this sampling distribution? c. What would be the standard deviation of the sampling distribution?

31\. Mislabeled seafood In 2013 the environmental group Oceana (usa.oceana.org) analyzed 1215 samples of seafood purchased across the United States and genetically compared the pieces to standard gene fragments that can identify the species. Laboratory results indicated that \(33 \%\) of the seafood was mislabeled according to U.S. Food and Drug Administration guidelines. a. Construct a \(95 \%\) confidence interval for the proportion of all seafood sold in the United States that is mislabeled or misidentified. b. Explain what your confidence interval says about seafood sold in the United States. c. A 2009 report by the Government Accountability Office says that the Food and Drug Administration has spent very little time recently looking for seafood fraud. Suppose an official said, "That's only 1215 packages out of the billions of pieces of seafood sold in a year. With the small number tested, I don't know that one would want to change one's buying habits." (An official was quoted similarly in a different but similar context). Is this argument valid? Explain.

Death penalty, again In the survey on the death penalty you read about in the Step-by-Step Example, the Gallup Poll actually split the sample at random, asking 510 respondents the question quoted earlier, "Generally speaking, do you believe the death penalty is applied fairly or unfairly in this country today?" The other 510 were asked, "Generally speaking, do you believe the death penalty is applied unfairly or fairly in this country today?" Seems like the same question, but sometimes the order of the choices matters. Suppose that for the second way of phrasing it, \(64 \%\) said they thought the death penalty was fairly applied. (Recall that \(53 \%\) of the original 510 thought the same thing.) a. What kind of bias may be present here? b. If we combine them, considering the overall group to be one larger random sample of 1020 respondents, what is a \(95 \%\) confidence interval for the proportion of the general public that thinks the death penalty is being fairly applied? c. How does the margin of error based on this pooled sample compare with the margins of error from the separate groups? Why?
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