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Chapter 9: Problem 94

State the null hypothesis, \(H_{o},\) and the alternative hypothesis, \(H_{a},\) that would be used to test these claims: a. The probability of our team winning tonight is less than 0.50. b. At least \(50 \%\) of all parents believe in spanking their children when appropriate. c. At most, \(80 \%\) of the invited guests will attend the wedding. d. The single-digit numbers generated by the computer do not seem to be equally likely with regard to being odd or even. e. Less than half of the customers like the new pizza.

Short Answer

Expert verified
a. \(H_o\): 'The probability of our team winning tonight is not less than 0.50', \(H_a\): 'The probability of our team winning tonight is less than 0.50.'\nb. \(H_o\): 'Less than 50% of all parents believe in spanking their children when appropriate.', \(H_a\): 'At least 50% of all parents believe in spanking their children when appropriate.'\nc. \(H_o\): 'More than 80% of the invited guests will attend the wedding.', \(H_a\): 'At most, 80% of the invited guests will attend the wedding.'\nd. \(H_o\): 'The single-digit numbers generated by the computer are equally likely to be odd or even.', \(H_a\): 'The single-digit numbers generated by the computer are not equally likely to be odd or even.'\ne. \(H_o\): 'At least half of the customers like the new pizza.', \(H_a\): 'Less than half of the customers like the new pizza.'

Step by step solution

01

Define the Hypotheses for Statement a.

Statement a states: 'The probability of our team winning tonight is less than 0.50.' The null hypothesis, \(H_o\), is usually the status quo or what is currently believed to be true, so in this case \(H_o\): 'The probability of our team winning tonight is not less than 0.50'. The alternative hypothesis, \(H_a\), is what we are attempting to prove, so in this case \(H_a\): 'The probability of our team winning tonight is less than 0.50.'
02

Define the Hypotheses for Statement b.

Statement b states: 'At least 50% of all parents believe in spanking their children when appropriate.' Therefore, the null hypothesis, \(H_o\), is: 'Less than 50% of all parents believe in spanking their children when appropriate.' The alternative hypothesis, \(H_a\), is the original statement: 'At least 50% of all parents believe in spanking their children when appropriate.'
03

Define the Hypotheses for Statement c.

Statement c states: 'At most, 80% of the invited guests will attend the wedding.' The null hypothesis, \(H_o\), would be: 'More than 80% of the invited guests will attend the wedding.' The alternative hypothesis, \(H_a\), would be the actual statement: 'At most, 80% of the invited guests will attend the wedding.'
04

Define the Hypotheses for Statement d.

Statement d states: 'The single-digit numbers generated by the computer do not seem to be equally likely with regard to being odd or even.' So, the null hypothesis, \(H_o\), would be: 'The single-digit numbers generated by the computer are equally likely to be odd or even.' The alternative hypothesis, \(H_a\), would be the original statement: 'The single-digit numbers generated by the computer are not equally likely to be odd or even.'
05

Define the Hypotheses for Statement e.

Statement e states: 'Less than half of the customers like the new pizza.' The null hypothesis, \(H_o\), would be: 'At least half of the customers like the new pizza.' The alternative hypothesis, \(H_a\), would be the original statement: 'Less than half of the customers like the new pizza.'

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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 foundational in hypothesis testing. It represents the default position or the conventional belief about a parameter and assumes no effect or no difference. It is often symbolized as \( H_o \). When you perform hypothesis testing, the null hypothesis is what you initially assume to be true.
It's like a starting point for research, where no change or new effect is expected.
  • For a hypothesis test, if the null hypothesis is rejected, it implies that the data provide enough evidence against it, suggesting that the alternative hypothesis might be true.
  • The null hypothesis often contains an equal sign (\( = \), \( \leq \), \( \geq \)), indicating that there is no change or effect.
Throughout the solutions, for each statement, the null hypothesis was set up to reflect this baseline assumption. For instance, statement e's null hypothesis is "At least half of the customers like the new pizza," which assumes no decrease in preference.
Alternative Hypothesis
The "Alternative Hypothesis," denoted by \( H_a \), is the claim or statement that a researcher wants to prove or verify. It is set in direct contrast to the null hypothesis.The alternative hypothesis is crucial because it represents new findings and potential discoveries. If the evidence strongly suggests that the null hypothesis is incorrect, researchers may accept the alternative hypothesis.
  • It is typically what we want to test, such as proving an effect or difference exists.
  • Unlike the null hypothesis, it often contains directional signs (\(<\), \(>\), \(eq\)) to show deviations.
For each of the original exercise statements, the alternative hypothesis reflected the claim made. For statement b, "at least 50%" appears in \( H_a \), highlighting the effort to verify whether belief in spanking meets or exceeds this percentage.
Probability
Probability is a measure of the likelihood of an event occurring and is often expressed as a number between 0 and 1. It plays a vital role in hypothesis testing, as researchers use it to determine the chance of observing certain data under the null hypothesis. Understanding probability helps in assessing whether a claim is statistically significant or not. Statistical tests calculate probabilities (known as p-values) to evaluate how consistent the results are with the null hypothesis.
  • A lower probability (or p-value) under the null hypothesis means the observed result is less likely by chance, pointing towards rejecting the null hypothesis.
  • Probability values also aid in making claims about population parameters based on sample data.
Probability is crucial in the exercise scenarios, such as statement a about the team winning with less than 0.50 chance, underpinning the need for objective evaluation.
Statistical Claims
Statistical claims involve statements about populations or parameters based on sample data. Hypothesis testing is fundamental in evaluating these claims, providing a structured method to support or refute them. When making a statistical claim, researchers form hypotheses, collect data, and use statistical tests to determine whether the original claim can be supported with enough evidence.
  • Claims often involve language like "less than," "at least," or "at most," because they compare different possibilities or conditions.
  • Statistical claims can address beliefs, differences, or predictions about variables or conditions.
In the exercise, each statement is a statistical claim about the population, like the likelihood of people attending an event, necessitating careful framing of hypotheses.
Statistical Significance
Statistical significance is the measure of whether the results of a hypothesis test are strong enough to reject the null hypothesis. In essence, it tells us if the data provide solid evidence to support the alternative hypothesis.The significance level, often denoted by \( \alpha \), is the threshold for deciding whether a result is statistically significant. Common levels are 0.05, 0.01, and 0.10, indicating the probability of rejecting the null hypothesis when it is actually true.
  • If the p-value is less than the chosen \( \alpha \), the result is statistically significant, and the null hypothesis is rejected.
  • Statistical significance doesn't mean practical significance, so results should also be interpreted in context.
In hypothesis testing scenarios like those in the exercise, statistical significance helps determine the credibility of claims like the fairness of number generation or customer preferences.

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Most popular questions from this chapter

All tomatoes that a certain supermarket buys from growers must meet the store's specifications of a mean diameter of \(6.0 \mathrm{cm}\) and a standard deviation of no more than \(0.2 \mathrm{cm} .\) The supermarket's buyer visits a potential new supplier and selects a random sample of 36 tomatoes from the grower's greenhouse. The diameter of each tomato is measured, and the mean is found to be 5.94 and the standard deviation is \(0.24 .\) Do the tomatoes meet the supermarket's specs? a. Determine whether an assumption of normality is reasonable. Explain. b. Is the sample evidence sufficient to conclude that the tomatoes do not meet the specs with regard to the mean diameter? Use \(\alpha=0.05\). c. Is the sample evidence sufficient to conclude that the tomatoes do not meet the specs with regard to the standard deviation? Use \(\alpha=0.05\). d. Write a short report for the buyer outlining the findings and recommendations as to whether or not to use this tomato grower to supply tomatoes for sale in the supermarket.

Bright-Lite claims that its 60 -watt light bulb burns with a length of life that is approximately normally distributed with a standard deviation of 81 hours. A sample of 101 bulbs had a variance of \(8075 .\) Is this sufficient evidence to reject Bright-Lite's claim in favor of the alternative,"the standard deviation is larger than 81 hours," at the 0.05 level of significance?

Find the test statistic for the hypothesis test: a. \(H_{o}: \sigma^{2}=532\) versus \(H_{a}: \sigma^{2}>532\) using sample information \(n=18\) and \(s^{2}=785\) b. \(H_{o}: \sigma^{2}=52\) versus \(H_{a}: \sigma^{2} \neq 52\) using sample information \(n=41\) and \(s^{2}=78.2\)

Find \(\alpha,\) the area of one tail, and the confidence coefficients of \(z\) that are used with each of the following levels of confidence. a. \(1-\alpha=0.80\) b. \(1-\alpha=0.98\) c. \(1-\alpha=0.75\)

It is important that the force required to extract a cork from a wine bottle not have a large standard deviation. Years of production and testing indicate that the no.9 corks in Applied Example 6.13 (p. 285 ) have an extraction force that is normally distributed with a standard deviation of 36 Newtons. Recent changes in the manufacturing process are thought to have reduced the standard deviation. a. What would be the problem with the standard deviation being relatively large? What would be the advantage of a smaller standard deviation? A sample of 20 randomly selected bottles is used for testing. Extraction Force in Newtons $$\begin{array}{llllllllll}\hline 296 & 338 & 341 & 261 & 250 & 347 & 336 & 297 & 279 & 297 \\\259 & 334 & 281 & 284 & 279 & 266 & 300 & 305 & 310 & 253 \\\\\hline\end{array}$$ b. Is the preceding sample sufficient to show that the standard deviation of extraction force is less than 36.0 Newtons, at the 0.02 level of significance? During a different testing, a sample of eight bottles is randomly selected and tested. Extraction Force in Newtons $$\begin{array}{rrrrrr}331.9 & 312.0 & 289.4 & 303.6 & 346.9 & 308.1 & 346.9 & 276.0\end{array}$$ c. Is the preceding sample sufficient to show that the standard deviation of extraction force is less than 36.0 Newtons, at the 0.02 level of significance? d. What effect did the two different sample sizes have on the calculated test statistic in parts b and c? What effect did they have on the \(p\) -value or critical value? Explain. e. What effect did the two different sample standard deviations have on the answers in parts b and c? What effect did they have on the \(p\) -value or critical value? Explain.
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