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Chapter 8: Problem 22

A 25 -question multiple-choice questionnaire has four choices for each question. Suppose an applicant is an expert and knows all the correct answers. The employer carries out a hypothesis test to determine whether a job applicant was answering randomly. The null hypothesis is \(p=0.25\), where \(p\) is the probability of correct answer.

Short Answer

Expert verified
To solve this exercise, the calculation of expected value and standard deviation under null hypothesis are derived first. Afterwards, with obtained Z value and p-value, the hypothesis test results are determined by comparing p-value with the significance level.

Step by step solution

01

Identify the Probability Distribution

The problem describes a situation dealing with a number of trials (25 multiple-choice questions), with each trial having a constant probability of success, where success denotes identifying the correct answer. This indicates that the problem can be dealt with using the Binomial Distribution.
02

Determine Expected Value and Standard Deviation

Here, n (trials) is 25, and p (probability of success) is 0.25 under null hypothesis. The expected value (mean) and standard deviation of a binomial distribution are given by: \[E(X) = np\] and \[SD(X) = \sqrt{np(1-p)}\] Therefore, calculate expected value and standard deviation assuming null hypothesis is true.
03

Calculate Test Statistic and p-value

In hypothesis testing, once mean and standard deviation are obtained, the Z value (Z-Score) for the test statistic could be determined by: \[Z = \frac{X - np}{\sqrt{np(1-p)}}\] where X is total successes (correct answers by candidate). Calculate Z value for given X. Afterwards, calculate p-value, which is the probability of observing a statistic as extreme as the test statistic, assuming null hypothesis.
04

Determine Results of the Hypothesis Test

After calculating p-value, compare it to the significance level (typically 0.05). If the p-value is less than or equal to the significance level, it indicates that the results are statistically significant and null hypothesis can be rejected. If p-value is higher, the null hypothesis cannot be rejected without additional information.

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Key Concepts

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

Binomial Distribution
The Binomial Distribution is a probability distribution commonly used in statistics when an experiment consists of a fixed number of independent trials, each with only two possible outcomes: "success" or "failure." In the context of the given exercise, each question on the multiple-choice test is considered a trial, where choosing the correct answer represents a "success."
  • The number of trials is denoted by \( n \), which in this exercise is 25, corresponding to the 25 questions.
  • The probability of success for each trial is given by \( p \). Here, it is 0.25, representing the chance of selecting the correct answer by guessing randomly.
  • In a binomial experiment, the key assumptions are that each trial is independent and the probability of success remains constant across trials.
Using these parameters, the properties of the binomial distribution help in finding probabilities related to the number of successes, such as determining the likelihood of a test-taker pondering or guessing the answers.
Null Hypothesis
In hypothesis testing, the Null Hypothesis is a statement that assumes there is no significant effect or relationship in the population. We use it to challenge or test via statistical evidence. In this exercise, the null hypothesis is that the applicant selects the correct answer 25% of the time, i.e., \( p = 0.25 \).
  • Formulating a null hypothesis involves expressing a specific and testable prediction in mathematical terms.
  • The purpose is to evaluate whether any observed effect is due to chance.
  • A null hypothesis sets a reference point to compare the empirical data outcome against.
Through hypothesis testing, we compare the observed data with this assumed condition. If the data strongly contradicts the null hypothesis, we might consider rejecting it in favor of an alternative hypothesis, suggesting a significant deviation from the accepted belief.
p-value
The p-value is a critical component in hypothesis testing and is the measure used to determine the significance of the test results. It quantifies the probability of observing data at least as extreme as the results obtained, under the assumption that the null hypothesis is true.
  • P-value helps in assessing whether to reject or fail to reject the null hypothesis.
  • If the p-value is small (typically \( \leq 0.05 \)), it suggests that the observed pattern is unlikely to have occurred under the null hypothesis.
  • A p-value larger than the preset significance level implies insufficient evidence to reject it.
The lower the p-value, the stronger the evidence against the null hypothesis. This serves as a guide for making conclusions about the hypothesis.
Z-Score
The Z-Score, also known as the standard score, is a numerical measurement in hypothesis testing that describes a value's relation to the mean of a group of values. It represents the number of standard deviations an element is from the mean. In the context of the problem, we calculate the Z-Score to assess the job applicant's performance compared to the expected outcome under the null hypothesis.
  • The Z-Score is calculated using the formula: \( Z = \frac{X - np}{\sqrt{np(1-p)}} \)
  • \( X \) represents the number of successful outcomes (correctly answered questions).
  • This metric helps convert data to a standard form, allowing for comparison across different data sets.
Utilizing the Z-Score, we can evaluate how extraordinarily or unexpectedly a test-taker's performance differs based on assumed probabilities. A significant Z-Score suggests a potential deviation in typical performance, lending weight to rejecting or failing to reject the null hypothesis.

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Most popular questions from this chapter

A study is done to see whether a coin is biased. The alternative hypothesis used is two-sided, and the obtained \(z\) -value is 2 . Assuming that the sample size is sufficiently large and that the other conditions are also satisfied, use the Empirical Rule to approximate the p-value.

Refer to Exercise 8.3. Suppose 100 people attend boot camp and 44 of them return to prison within three years). The population recidivism rate for the whole state is \(40 \%\). a. What is \(\hat{p}\), the sample proportion of successes? (It is somewhat odd to call retuming to prison a success.) b. What is \(p_{0}\), the hypothetical proportion of success under the null hypothesis? c. What is the value of the test statistic? Explain in context.

When, in a criminal court, a defendant is found "not guilty," is the court saying with certainty that he or she is innocent? Explain.

Suppose a friend says he can predict whether a coin flip will result in heads or tails. You test him, and he gets 20 right out of \(20 .\) Do you think he can predict the coin flip (or has a way of cheating)? Or could this just be something that is likely to occur by chance? Explain without performing any calculations.

An arthritis diet claims that the disease can be relieved by reducing sugar from the diet. To test this claim, a researcher randomly assigns arthritis patients to two groups. Both groups eat the same amount of calories, but one group eats almost no sugar and the other group includes sugar in their meal. After 3 months, the doctor tests the claim that the sugar-free diet is better than the usual diet. She records the proportion of each group that got relieved of almost \(10 \%\) of their problem. She then announced that she failed to reject the null hypothesis. Which of the following are valid interpretations of her findings? a. The sugar-free dict was less effective than the normal diet. b. The researcher did not see enough evidence to conclude that the sugarfree diet was more effective. c. The sugar-free diet and the normal diet were equally effective. d. There were no significant differences in effectiveness between the sugar- free diet and normal diet.
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