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Chapter 8: Problem 36

A random survey showed that 1680 out of 2015 surveyed employees favored salary deduction for late attendance. a. Test the hypothesis that more than half of the employees favor salary deduction using a significance level of \(0.05 .\) Label each step. b. If there were a vote by the public about whether to discontinue the salary deduction, would it pass? (Base your answer on part a.)

Short Answer

Expert verified
The solution involves the testing of hypotheses to ascertain whether more than half of the employees are in favour of the salary deduction. By comparing our calculated Z-score with the critical value, we conclude that if the Z-score is greater, we'll reject the null hypothesis. This means that more than half of the employees favor the salary deduction. If a vote were to occur about whether to discontinue the salary deduction, based on the results of the hypothesis test, it would probably not pass.

Step by step solution

01

Formulating the hypotheses

The null hypothesis \( H_{0} \) is that half or less of the employees are in favor, i.e., \( p_{0} = 0.5 \). The Alternative hypothesis \( H_{1} \) is that more than half of the employees are in favor, i.e., \( p > 0.5 \). Therefore,\( H_{0}: p_{0} = 0.5 \),\( H_{1}: p > 0.5 \).
02

Calculating Z-score

The z-score is calculated using the formula \( Z = \frac{p - p_{0}}{\sqrt{\frac{p_{0}(1- p_{0})}{n}}} \), where \( p = \frac{1680}{2015} \) is the sample proportion, \( p_{0} = 0.5 \) is the population proportion and \( n = 2015 \) is the size of the sample. Substituting these values into the formula would give us the z-score.
03

Knowing the critical value

The critical value for a one-tailed test at 0.05 significance level is approximately 1.645. This value can be found from the z-table or from the calculator.
04

Making Decision

Here, we compare our calculated Z-score with the critical value. If the Z-score is greater than the critical value, we reject the null hypothesis. Otherwise, we do not reject the null hypothesis.
05

Interpreting the results

If we reject the null hypothesis, it implies that more than half of the employees favor salary deduction.
06

Predicting the vote

Based on the results of the hypothesis test, if it was found that more than half of the employees favor salary deduction, we can predict that if there were a vote by the public about whether to discontinue the salary deduction, it probably would not pass.

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Key Concepts

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

Null Hypothesis
Every hypothesis test begins with the formulation of hypotheses. When conducting a hypothesis test, the null hypothesis (\( H_0 \)) is the initial claim we make. In our example, the null hypothesis states that half or fewer of the employees favor a salary deduction for late attendance. To express this mathematically, we use the symbol for proportion:
  • If the null hypothesis is true, the proportion of employees in favor equals or is less than 0.5 (\( p_0 = 0.5 \)).
  • The null hypothesis serves as a baseline that we later challenge by gathering data and performing calculations. It is an essential part of statistical reasoning.
When formulating hypotheses, always ensure the null hypothesis is a statement of no change or no effect. This forms the basis for comparison against the alternative hypothesis.
Alternative Hypothesis
Once the null hypothesis is established, we define the alternative hypothesis (\( H_1 \)), which counters \( H_0 \). In this situation, the alternative hypothesis suggests that more than half of the employees favor a salary deduction for late attendance. Mathematically written as:
  • The proportion of employees favoring the deduction exceeds 0.5 (\( p > 0.5 \)).
  • Alternative hypotheses are tested with data. If data provides strong evidence against the null hypothesis, we may consider the alternative hypothesis to be a better explanation of the facts.
This hypothesis is supposed to show an existing effect or change, posing the real question we want to investigate.
Z-score
The z-score is a crucial element in hypothesis testing, allowing us to understand our sample result relative to the null hypothesis. It measures how many standard deviations our sample proportion is away from the null hypothesis proportion. In this exercise:
  • The formula used is \( Z = \frac{p - p_0}{\sqrt{\frac{p_0(1- p_0)}{n}}} \), where:
    • \( p \) is the sample proportion, calculated as \( \frac{1680}{2015} \)
    • \( p_0 = 0.5 \), is the proportion under the null hypothesis
    • \( n = 2015 \) is the sample size
  • The computed z-score communicates how 'unusual' the sample results are, under the assumption that the null hypothesis is true.
A high absolute z-score value suggests that the observed data is less likely under the null hypothesis, moving us closer to rejecting it.
Significance Level
In hypothesis testing, the significance level (\( \alpha \)) is a threshold set by the researcher to determine how strong the evidence must be to reject the null hypothesis. For our case:
  • A significance level of \( 0.05 \) is chosen, which implies a 5% risk of rejecting the null hypothesis when it is actually true.
  • This level affects how critical values are determined. For example, for a one-tailed test:
    • The critical z-value is approximately 1.645, found from z-tables or statistical tools.
  • If the computed z-score exceeds this critical value, it suggests that the evidence in favor of the alternative hypothesis is strong enough to reject the null hypothesis.
Essentially, the choice of significance level reflects the precision we require in our judgment about the hypotheses, guiding decision-making in statistical analysis.

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Most popular questions from this chapter

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.

A coin is flipped 30 times and comes up heads 18 times. You want to test the hypothesis that the coin does not come up \(50 \%\) heads in the long run. Pick the correct null hypothesis for this test. i. \(\mathrm{H}_{0}: \hat{p}=0.60\) ii. \(\mathrm{H}_{0}: p=0.50\) iii. \(\mathrm{H}_{0}: p=0.60\) iv. \(\mathrm{H}_{0}=\hat{p}=0.50\)

Harry was tested to see whether he could tell cotton fabric from rayon fabric by touch alone. He got 18 instances right out of 25, and the p-value was \(0.174 .\) Explain what is wrong with the following inference, and write a correct inference: "We proved that Harry cannot tell the difference between cotton and rayon fabrics by touch."

Many polls have asked people whether they are trying to lose weight. A Gallup Poll in November of 2008 showed that \(22 \%\) of men said they were seriously trying to lose weight. In \(2006,24 \%\) of men (with the same average weight of 194 pounds as the men polled in 2008 ) said they were seriously trying to lose weight. Assume that both samples contained 500 men. a. Determine how many men in the sample from 2008 and how many in the sample from 2006 said they were seriously trying to lose weight. b. Determine whether the difference in proportions is significant at the \(0.05\) level. c. Repeat the problem with the same proportions but assuming both sample sizes are now 5000 . d. Comment on the different p-values and conclusions with different sample sizes.

A proponent of a new proposition on a ballot wants to know whether the proposition is likely to pass. Suppose a poll is taken, and 580 out of 1000 randomly selected people support the proposition. Should the proponent use a hypothesis test or a confidence interval to answer this question? Explain. If it is a hypothesis test, state the hypotheses and find the test statistic, p-value, and conclusion. If a confidence interval is appropriate, find the approximate \(95 \%\) confidence interval. In both cases, assume that the necessary conditions have been met.
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