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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

Are boys more likely? We hear that newborn babies are more likely to be boys than girls. Is this true? \(\mathrm{A}\) random sample of 25,468 firstborn children included 13,173 boys. (a) Do these data give convincing evidence that firstborn children are more likely to be boys than girls? (b) To what population can the results of this study be generalized: all children or all firstborn children? Justify your answer.

A college president says, "99\% of the alumni support my firing of Coach Boggs." You contact an SRS of 200 of the college's 15,000 living alumni to perform a test of \(H_{0}: p=0.99\) versus \(H_{a}: p<0.99\)

Vigorous exercise helps people live several years longer (on average). Whether mild activities like slow walking extend life is not clear. Suppose that the added life expectancy from regular slow walking is just 2 months. A statistical test is more likely to find a significant increase in mean life expectancy if (a) it is based on a very large random sample and a \(5 \%\) significance level is used. (b) it is based on a very large random sample and a \(1 \%\) significance level is used. (c) it is based on a very small random sample and a \(5 \%\) significance level is used. (d) it is based on a very small random sample and a \(1 \%\) significance level is used. (e) the size of the sample doesn't have any effect on the significance of the test.

Researchers suspect that Variety A tomato plants have a higher average yield than Variety \(\mathrm{B}\) tomato plants. To find out, researchers randomly select 10 Variety A and 10 Variety \(\mathrm{B}\) tomato plants. Then the researchers divide in half each of 10 small plots of land in different locations. For each plot, a coin toss determines which half of the plot gets a Variety A plant; a Variety \(\mathrm{B}\) plant goes in the other half. After harvest, they compare the yield in pounds for the plants at each location. The 10 differences in yield (Variety \(\mathrm{A}\) - Variety \(\mathrm{B}\) ) are recorded. A graph of the differences looks roughly symmetric and single-peaked with no outliers. A paired \(t\) test on the differences yields \(t=1.295\) and \(P\) -value \(=0.1138\). (a) State appropriate hypotheses for the paired \(t\) test. Be sure to define your parameter. (b) What are the degrees of freedom for the paired \(t\) test? (c) Interpret the \(P\) -value in context. What conclusion should the researchers draw? (d) Describe a Type I error and a Type II error in this setting. Which mistake could researchers have made based on your answer to part (c)?

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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