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Chapter 8: Problem 5

A biased lotto draws even numbers faster than odd numbers. A token is drawn 50 times, and 35 even numbers come up. a. Pick the correct null hypothesis: i. \(\hat{p}=0.50\) ii. \(\hat{p}=0.70\) iii. \(p=0.50\) iv. \(p=0.70\) b. Pick the correct alternative hypothesis: i. \(\hat{p}=0.50\) ii. \(\hat{p}=0.70\) iii. \(p>0.50\) iv. \(p>0.70\)

Short Answer

Expert verified
The correct null hypothesis is \( p = 0.50 \) and the correct alternative hypothesis is \( p > 0.50 \)

Step by step solution

01

Understanding Null Hypothesis

The lotto seems to draw even numbers more frequently than odd numbers. Let \( p \) be the proportion of times an even number is drawn. \n\nWe have to assume an initial state where there is no bias in the number drawing process; meaning, the process equally draws even and odd numbers. So, the initial assumption is that half the numbers drawn are even. This state is considered as the null hypothesis. Thus, for this problem \( p = 0.50 \) is the correct null hypothesis. So, the right option is iii. \( p = 0.5 \).
02

Understanding Alternative Hypothesis

Considering that the null hypothesis is \( p = 0.50 \) (that is, there is no difference in outcomes), the alternative hypothesis would be that there is a difference.\n\nIn this case, since the biased lottery is conjectured to draw even numbers more than 50% of the time, the alternative hypothesis would be \( p > 0.50 \). So, the correct alternative hypothesis is iii. \( p > 0.50 \).

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Key Concepts

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

Null Hypothesis
In hypothesis testing, the "null hypothesis" represents a statement of no effect or no difference. It's a way of stating that whatever condition you're testing doesn't actually cause any change.

In the example of a lottery drawing, we use the null hypothesis to say that even numbers are drawn just as often as odd numbers. This assumption helps us begin our test without any prior judgment about which numbers are drawn more often.
  • Formally, the null hypothesis can be represented by a statement like "the proportion of even numbers drawn is 0.50," meaning 50% of the time, an even number is drawn just as would be expected by pure chance. If so, it can be expressed as: \[p = 0.50 \]
  • The null hypothesis works as a starting point for statistical testing. Its goal is to remain neutral until the data suggests otherwise.
  • Rejecting the null hypothesis indicates that there is significant evidence that supports a different outcome.
Selecting the correct null hypothesis is crucial because it provides the baseline from which changes or effects are measured.
Alternative Hypothesis
The "alternative hypothesis" in hypothesis testing suggests that there is an effect or a difference. It posits what we might believe if the evidence rejects the null hypothesis.

In the context of the biased lottery, where even numbers seem to be drawn more frequently than odd numbers, the alternative hypothesis goes against the assumption of equal chance.
  • While the null hypothesis assumes no bias, the alternative hypothesis there suggests that the proportions favor even numbers. This makes us say: \[p > 0.50\]
  • Essentially, it claims that the proportion of even numbers drawn is greater than 50%.
  • The alternative hypothesis offers a potential higher truth to what might be occurring in the system being tested; it motivates the search for proof.
If statistical tests show enough evidence to support the alternative hypothesis, it helps us affirmatively state that the lottery indeed favors even numbers.
Proportion
In statistics, a proportion refers to a part or fraction of a whole, usually expressed as a percentage or a ratio. When involved in hypothesis testing, proportion is used to represent probabilities of certain events occurring.

In this exercise, proportion helps us analyze the likelihood of drawing even numbers in a biased lotto situation.
  • Initially, we set the proportion, denoted by \( p \), as 0.50 for the null hypothesis, meaning the chances of drawing an even number equals that of drawing an odd number.
  • In calculating the alternative hypothesis, if we suspect a bias, we then compare the actual observed proportion to what would be expected by chance.
  • The actual observed proportion is based on the data collected, such as observing 35 even draws out of 50 total draws. This can be represented as: \[\hat{p} = \frac{35}{50} = 0.70\]
Understanding proportion allows us to quantitatively evaluate claims of bias or deviation in the lottery system. This measure is fundamental for hypothesis testing, as it helps compare observed outcomes versus expected probabilities.

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Most popular questions from this chapter

The Gallup organization frequently conducts polls in which they ask the following question: "In general, do you feel that the laws covering the sale of firearms should be made more strict, less strict, or kept as they are now?" In February \(1999,60 \%\) of those surveyed said "more strict," and on April 26,1999, shortly after the Columbine High School shootings, \(66 \%\) of those surveyed said "more strict." a. Assume that both polls used samples of 560 people. Determine the number of people in the sample who said "more strict" in February 1999 , before the school shootings, and the number who said "more strict" in late April 1999 , after the school shootings. b. Do a test to see whether the proportion that said "more strict" is statistically significantly different in the two different surveys, using a significance level of \(0.01\). c. Repeat the problem, assuming that the sample sizes were both 1120 . d. Comment on the effect of different sample sizes on the \(\mathrm{p}\) -value and on the conclusion.

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.)

Standard anticoagulant therapy (to prevent blood clots) requires frequent laboratory monitoring to prevent internal bleeding. A new procedure using rivaroxaban (riva) was tested because it does not require frequent monitoring. A randomized trial (Einstein-PE Investigators 2012 ) was carried out, with standard therapy being randomly assigned to half of 4832 patients and riva randomly assigned to the other half. A bad result was recurrence of a blood clot in a vein. Fifty of the 2416 patients on standard therapy had a bad outcome, and 44 of the 2416 patients on riva had a bad outcome. a. Test the hypothesis that the proportions of bad results are different for riva and standard therapy patients. Use a significance level of \(0.05\), and show all four steps. b. Using methods leamed in Chapter 7 , estimate the difference between the two population proportions using a \(95 \%\) confidence interval, and comment on how it can be used to evaluate the null hypothesis in part a.

A study is done to see whether a coin is biased. The alternative hypothesis used is two-sided, and the obtained \(z\) -value is 1 . Assuming that the sample size is sufficiently large and that the other conditions are also satisfied, use the Empirical Rule to approximate the \(\mathrm{p}\) -value.

Literacy in 2015 In March 2016, the UNESCO Institute for Statistics (UIS) reported that the literacy rate in Zimbabwe was \(88.5 \%\) for males and \(84.6 \%\) for females. Would it be appropriate to draw a two-proportion z-test to determine whether the rates for males and females were significantly different (assuming we knew the total number of males and females)? Explain.
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