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Chapter 10: Problem 43

Students at the Akademia Podlaka conducted an experiment to determine whether the Belgium-minted Euro coin was equally likely to land heads up or tails up. Coins were spun on a smooth surface, and in 250 spins, 140 landed with the heads side up (New Scientist, January 4 , 2002). Should the students interpret this result as convincing evidence that the proportion of the time the coin would land heads up is not .5? Test the relevant hypotheses using \(\alpha=.01\). Would your conclusion be different if a significance level of \(.05\) had been used? Explain.

Short Answer

Expert verified
The short answer will depend on the calculated P-value and its comparison with the provided significance levels. It will be of the form: 'For \(\alpha = 0.01\), the evidence is insufficient (or sufficient) to say the coin is unfair. However, for \(\alpha = 0.05\), we reject (or do not reject) the null hypothesis indicating the coin is (or isn't) unfair.'

Step by step solution

01

State the null and alternate hypotheses

The Null Hypothesis Ho : p = 0.5 (The coin is fair - probability of landing heads is 0.5). The Alternate Hypotheses Ha : p 鈮� 0.5 (The coin is not fair - probability of landing heads is not 0.5)
02

Determine the Test Statistics

The test statistic for a hypothesis test for a proportion is a z-score (z). The formula for the z- score is \[ z = (x - np_0) / \sqrt{ np_0(1 - p_0)} \] where n is the number of trials, \(p_0\) is the expected probability under the null hypothesis and x is the number of successes. Here, n = 250 (number of spins), \(p_0 = 0.5\) and x = 140 (coins landing heads). Plugging these in the equation will give the z score.
03

Calculate the P-value

The P-value is the probability that a z-score is as extreme as, or more extreme than, the observed z-score, assuming the null hypothesis is true. It can be found using the standard normal distribution (z-distribution) and statistical tables or software. It is calculated for both sides of the distribution (because the alternate hypothesis is p 鈮� 0.5), hence it is twice the probability of the tail beyond the test statistic.
04

Compare the P-value to the predefined significance level

Compare the P-value with the provided significance levels (\(\alpha = 0.01\) and \( \alpha = 0.05\)). If the P-value is less than or equal to \(\alpha\), reject the null hypothesis. In this case, conclude the coin is not fair.
05

Interpret the result

There are two conclusions to be reached - one for each significance level. If for \(\alpha = 0.01\), the null hypothesis is not rejected, there is insufficient evidence to conclude the coin is unfair based on this level. If for \(\alpha = 0.05\), the null hypothesis is rejected, conclude the coin is unfair based on this level. The conclusion may change based on the chosen significance level.

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Key Concepts

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

Null Hypothesis
The concept of the null hypothesis is fundamental in statistics and hypothesis testing. In essence, it is a statement that there is no effect or no difference, and it represents a default or starting assumption.
The null hypothesis is denoted as \( H_0 \). In the context of the coin flipping exercise, the null hypothesis is that the coin is fair, meaning that the probability \( p \) of landing heads is 0.5.
This hypothesis serves as a baseline for comparison, against which statistical evidence is evaluated. The hypothesis is tested to be either rejected or not rejected based on the data collected. Often, the null hypothesis is assumed true until evidence suggests otherwise.
Significance Level
The significance level, denoted as \( \alpha \), is a threshold used to decide whether to reject the null hypothesis. It quantifies the probability of making a Type I error, which occurs when we wrongly reject a true null hypothesis.
Common significance levels are 0.05 and 0.01. In the problem from Akademia Podlaka, two significance levels are examined: \( \alpha = 0.01 \) and \( \alpha = 0.05 \).
The choice of significance level is crucial as it reflects how strong the evidence must be before we can reject the null hypothesis. A smaller \( \alpha \) means more stringent criteria for rejecting the null hypothesis.
P-Value
The P-value in hypothesis testing indicates the probability of obtaining a result that is as extreme as, or more extreme than, what was observed if the null hypothesis is true.
A small P-value suggests that the observed data is unlikely under the null hypothesis, providing evidence against it. In statistical tests, we compare the P-value to the significance level \( \alpha \).
If the P-value is less than or equal to \( \alpha \), we reject the null hypothesis. Conversely, if it is greater, we do not reject \( H_0 \). Thus, the P-value acts as a measure of evidence against the null hypothesis.
Z-Score
The Z-score is a statistical measure that describes a value's relation to the mean of a group of values. It's expressed in terms of standard deviations away from the mean.
In hypothesis testing, the Z-score helps determine how far away our sample statistic is from the expected value under the null hypothesis. The formula for calculating the Z-score in the context of proportions is:\[ z = \frac{x - np_0}{\sqrt{np_0(1 - p_0)}}\]where \( x \) is the observed number of successes, \( n \) is the number of trials, and \( p_0 \) is the probability under the null hypothesis.
The Z-score allows us to find the P-value, helping us decide whether to reject the null hypothesis.
Statistical Experiment
A statistical experiment is a process by which we collect data to analyze hypotheses. It involves defining hypotheses, collecting data, and using statistical methods to draw conclusions.
In the problem presented, the experiment involved spinning a coin 250 times and counting the number of heads. This setup allows us to apply statistical techniques to analyze whether the coin is biased.
Statistical experiments are designed to be repeatable and controlled, minimizing variables that could affect the outcome. They provide the foundation for hypothesis testing by ensuring that data is collected systematically.

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Most popular questions from this chapter

In a national survey of 2013 adults, 1590 responded that lack of respect and courtesy in American society is a serious problem, and 1283 indicated that they believe that rudeness is a more serious problem than in past years (Associated Press, April 3,2002 ). Is there convincing evidence that more than three-quarters of U.S. adults believe that rudeness is a worsening problem? Test the relevant hypotheses using a significance level of \(.05\).

Researchers have postulated that because of differences in diet, Japanese children have a lower mean blood cholesterol level than U.S. children do. Suppose that the mean level for U.S. children is known to be 170 . Let \(\mu\) represent the true mean blood cholesterol level for Japanese children. What hypotheses should the researchers test?

Past experience has indicated that the true response rate is \(40 \%\) when individuals are approached with a request to fill out and return a particular questionnaire in a stamped and addressed envelope. An investigator believes that if the person distributing the questionnaire is stigmatized in some obvious way, potential respondents would feel sorry for the distributor and thus tend to respond at a rate higher than \(40 \%\). To investigate this theory, a distributor is fitted with an eye patch. Of the 200 questionnaires distributed by this individual, 109 were returned. Does this strongly suggest that the response rate in this situation exceeds the rate in the past? State and test the appropriate hypotheses at significance level . 05 .

Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most \(150^{\circ} \mathrm{F}\), there will be no negative effects on the river ecosystem. To investigate whether the plant is in compliance with regulations that prohibit a mean discharge water temperature above \(150^{\circ} \mathrm{F}\), a scientist will take 50 water samples at randomly selected times and will record the water temperature of each sample. She will then use a \(z\) statistic $$ z=\frac{\bar{x}-150}{\frac{\sigma}{\sqrt{n}}} $$ to decide between the hypotheses \(H_{0}: \mu=150\) and \(H_{a^{2}}\) \(\mu>150\), where \(\mu\) is the mean temperature of discharged water. Assume that \(\sigma\) is known to be 10 . a. Explain why use of the \(z\) statistic is appropriate in this setting. b. Describe Type I and Type II errors in this context. c. The rejection of \(H_{0}\) when \(z \geq 1.8\) corresponds to what value of \(\alpha ?\) (That is, what is the area under the \(z\) curve to the right of \(1.8 ?\) ) d. Suppose that the true value for \(\mu\) is 153 and that \(H_{0}\) is to be rejected if \(z \geq 1.8 .\) Draw a sketch (similar to that of Figure \(10.5\) ) of the sampling distribution of \(\bar{x}\), and shade the region that would represent \(\beta\), the probability of making a Type II error. e. For the hypotheses and test procedure described, compute the value of \(\beta\) when \(\mu=153\). f. For the hypotheses and test procedure described, what is the value of \(\beta\) if \(\mu=160 ?\) g. If \(H_{0}\) is rejected when \(z \geq 1.8\) and \(\bar{x}=152.8\), what is the appropriate conclusion? What type of error might have been made in reaching this conclusion?

In a representative sample of 1000 adult Americans, only 430 could name at least one justice who is currently serving on the U.S. Supreme Court (Ipsos, January 10,2006 ). Using a significance level of \(.01\), carry out ? hypothesis test to determine if there is convincing evidence to support the claim that fewer than half of adult Americans can name at least one justice currently serving on the Supreme Court.
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