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Chapter 9: Problem 60

Which of the following is not a condition for performing a significance test about a population proportion \(p ?\) (a) The data should come from a random sample or randomized experiment. (b) Both \(n p_{0}\) and \(n\left(1-p_{0}\right)\) should be at least 10 . (c) If you are sampling without replacement from a finite population, then you should sample no more than \(10 \%\) of the population. (d) The population distribution should be approximately Normal, unless the sample size is large. (e) All of the above are conditions for performing a significance test about a population proportion.

Short Answer

Expert verified
Option (d) is not a condition for performing a significance test about a population proportion.

Step by step solution

01

Understand the Question

We need to identify which option is not a condition for performing a significance test about a population proportion \(p\). A significance test for a population proportion requires certain assumptions or conditions to ensure the results' validity.
02

Review Common Conditions for Significance Tests

For a significance test about a population proportion, common conditions include:1. The data must be from a random sample or a randomized experiment (Option a).2. For a normal approximation, both \(np_0\) and \(n(1-p_0)\) should be at least 10 (Option b).3. If sampling without replacement, the sample should be no more than 10% of the population (Option c).
03

Evaluate Option (d)

Option (d) states that the population distribution should be approximately Normal unless the sample size is large. However, this is not a standard condition for testing a population proportion. Population proportion tests do not typically require the population distribution to be Normal; rather, the sample data needs to meet the Normal approximation condition mentioned in Option (b).
04

Confirm the Answer

Since Option (d) is not a valid condition for a significance test about a population proportion, it is the answer. Other conditions (a, b, c) are standard requirements for ensuring the validity of the test.

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Key Concepts

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

Random Sampling Condition
When conducting a significance test for a population proportion, it is vital for the data to derive from a random sample or a randomized experiment. This condition ensures that the sample accurately represents the population, reducing sampling bias.
  • Random samples model the larger population accurately.
  • This accuracy means that the results are more reliable and can be generalized to the population.
  • Randomized experiments allow researchers to control variables and reduce the impact of confounding factors.
Meeting this condition forms the backbone of a trustworthy statistical analysis, making any inferences derived from the test dependable.
Normal Approximation
Normal approximation is a key aspect when testing a population proportion. This approximation technique allows us to use the Normal distribution to model the sampling distribution of the sample proportion. The primary condition for using this method is ensuring that both the expected number of successes and failures are at least 10. In mathematical terms, this is expressed as:
  • \( np_0 \geq 10 \)
  • \( n(1-p_0) \geq 10 \)
Here, \( n \) represents the sample size and \( p_0 \) the hypothesized population proportion.Meeting this criterion allows the sampling distribution to closely resemble a Normal distribution. This resemblance simplifies calculations and probability assessments using standard Normal distribution tables, streamlining the hypothesis testing process.
Sample Size Guidelines
Another important consideration when performing a significance test for population proportion is adhering to sample size guidelines. Particularly, when sampling without replacement from a finite population, the sample size should not exceed 10% of the total population. This guideline, known as the "10% Condition," helps maintain the independence of observations, which is crucial for accurate statistical inference.
  • Sampling a small fraction of the population ensures that the sample behaves more like a truly random sample.
  • Going beyond 10% may disrupt the natural variability, as the observations start becoming dependent on each other.
By maintaining a sample size within these boundaries, you ensure that the assumptions of the significance test hold, leading to more valid and reliable conclusions.

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Most popular questions from this chapter

A student performs a test of \(H_{0}: p=0.75\) versus \(H_{a}: p>0.75\) and gets a \(P\) -value of \(0.99 .\) The student writes: "Because the \(P\) -value is greater than \(0.75,\) we reject \(H_{0} .\) The data prove that \(H_{a}\) is true." Explain what is wrong with this conclusion.

After once again losing a football game to the archrival, a college's alumni association conducted a survey to see if alumni were in favor of firing the coach. An SRS of 100 alumni from the population of all living alumni was taken, and 64 of the alumni in the sample were in favor of firing the coach. Suppose you wish to see if a majority of living alumni are in favor of firing the coach. The appropriate test statistic is (a) \(z=\frac{0.64-0.5}{\sqrt{\frac{0.64(0.36)}{100}}}\) (b) \(t=\frac{0.64-0.5}{\sqrt{\frac{0.64(0.36)}{100}}}\) (c) \(z=\frac{0.64-0.5}{\sqrt{\frac{0.5(0.5)}{100}}}\) (d) \(z=\frac{0.64-0.5}{\sqrt{\frac{0.64(0.36)}{64}}}\) (e) \(z=\frac{0.5-0.64}{\sqrt{\frac{0.5(0.5)}{100}}}\)

For the study of Jordanian children in Exercise 2 , the sample mean hemoglobin level was \(11.3 \mathrm{~g} / \mathrm{dl}\) and the sample standard deviation was \(1.6 \mathrm{~g} / \mathrm{dl} .\) A significance test yields a \(P\) -value of 0.0016 . (a) Explain what it would mean for the null hypothesis to be true in this setting. (b) Interpret the \(P\) -value in context.

The French naturalist Count Buffon \((1707-1788)\) tossed a coin 4040 times. He got 2048 heads. That's a bit more than one-half. Is this evidence that Count Buffon's coin was not balanced? To find out, Luisa decides to perform a significance test. Unfortunately, she made a few errors along the way. Your job is to spot the mistakes and correct them. $$ \begin{array}{l} H_{0}: \mu>0.5 \\ H_{a}: \bar{x}=0.5 \end{array} $$ \(\bullet\quad\) \(10 \%: 4040(0.5)=2020\) and \(4040(1-0.5)=2020\) are both at least 10 . \(\bullet\quad\) Large Counts: There are at least 40,400 coins in the world. \(t=\frac{0.5-0.507}{\sqrt{\frac{0.5(0.5)}{4040}}}=-0.89 ; P\) -value \(=1-0.1867=0.8133\) Reject \(H_{0}\) because the \(P\) -value is so large and conclude that the coin is fair.

Does Friday the 13 th have an effect on people's behavior? Researchers collected data on the number of shoppers at a sample of 45 nearby grocery stores on Friday the 6 th and Friday the 1 3th in the same month. The dotplot and computer output below summarize the data on the difference in the number of shoppers at each store on these two days (subtracting in the order 6 th minus 13 th \() .^{25}\) Researchers would like to carry out a test of \(H_{0}: \mu_{d}=0\) versus \(H_{a}: \mu_{d} \neq 0,\) where \(\mu_{d}\) is the true mean difference in the number of grocery shoppers on these two days. Which of the following conditions for performing a paired \(t\) test are clearly satisfied? I. Random II. \(10 \%\) III. Normal/Large Sample (a) I only (b) II only (c) III only (d) I and II only (e) I, II, and III
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