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Chapter 4: Problem 147

In Exercises 4.146 to \(4.149,\) hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample and give the significance level used. Hypotheses: \(H_{0}: p=0.5\) vs \(H_{a}: p \neq 0.5\) (a) \(95 \%\) confidence interval for \(p: \quad 0.53\) to 0.57 (b) \(95 \%\) confidence interval for \(p: \quad 0.41\) to 0.52 (c) 99\% confidence interval for \(p: \quad 0.35\) to 0.55

Short Answer

Expert verified
For sample A, the null hypothesis that \(p=0.5\) is rejected with a 5% significance level. For samples B and C, the null hypothesis that \(p=0.5\) cannot be rejected with a 5% and 1% significance level, respectively.

Step by step solution

01

Analyze Confidence Interval for Sample A

The 95% confidence interval for sample A is between 0.53 and 0.57. Since the value from the null hypothesis (0.5) does not fall within this interval, the null hypothesis \(H_{0}: p=0.5\) is rejected for sample A with a 5% significance level.
02

Analyze Confidence Interval for Sample B

The 95% confidence interval for sample B is between 0.41 and 0.52. Since the value from the null hypothesis (0.5) does fall within this interval, the null hypothesis \(H_{0}: p=0.5\) cannot be rejected for sample B with a 5% significance level.
03

Analyze Confidence Interval for Sample C

The 99% confidence interval for sample C is between 0.35 and 0.55. Since the value from the null hypothesis (0.5) does fall within this interval, the null hypothesis \(H_{0}: p=0.5\) cannot be rejected for sample C with a 1% significance level.

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Key Concepts

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

Confidence Intervals
Understanding confidence intervals is fundamental in the realm of statistics, as they offer a range of plausible values for a population parameter (like a mean or proportion). A confidence interval (CI) tells us that we can be certain to some degree - typically 95% or 99% - that the true parameter falls within this range.

Let's consider the example from the exercise where hypotheses are tested using confidence intervals for different samples. Here, CIs are used to decide whether to reject the null hypothesis. If the interval does not contain the null hypothesis value, the hypothesis is rejected, assuming that our sample is representative of the population.

For instance, a 95% CI of 0.53 to 0.57 for a population proportion means that we can be 95% confident that the true proportion lies within this range. When the null hypothesis value of 0.5 is not within this interval, it allows us to conclude that there is a significant difference from the hypothesized value at a 5% significance level.
Null Hypothesis
A null hypothesis, denoted by H0, is a statement used in statistics that there is no effect or no difference, and it serves as the starting point for any statistical hypothesis testing. In hypothesis testing, it's the hypothesis that researchers aim to test against the alternative hypothesis, denoted by Ha or H1, which suggests that there is an effect or a difference.

In the given exercise, the null hypothesis is that the population proportion (p) is equal to 0.5, expressed as H0: p=0.5. It is used as a standard to measure against the provided confidence intervals. If the confidence interval includes the value of 0.5, we do not have sufficient evidence to reject the null hypothesis. Conversely, if the value of 0.5 does not reside within the interval, it implies that there might be a significant difference and leads to rejecting the null hypothesis.
Significance Level
The significance level, often denoted by alpha (伪), is a threshold chosen by the researcher that determines when to reject the null hypothesis. It is the probability of rejecting the null hypothesis when in fact it is true, also known as a Type I error.

Common levels include 0.05 (5%) and 0.01 (1%), corresponding to confidence levels of 95% and 99%, respectively. In our textbook example, a 95% confidence interval corresponds to a significance level of 0.05. In other words, if the confidence interval were to be calculated from numerous samples, the true proportion would fall outside of the interval 5% of the time. When an observed statistic falls outside the chosen confidence interval, we say it is 'statistically significant' at the given significance level.

To decide whether to reject the null hypothesis, one compares the significance level to the p-value of the test. If the p-value is less than or equal to the significance level, the null hypothesis is rejected, indicating significant evidence against it.

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Most popular questions from this chapter

Exercises 4.117 to 4.122 give null and alternative hypotheses for a population proportion, as well as sample results. Use StatKey or other technology to generate a randomization distribution and calculate a p-value. StatKey tip: Use "Test for a Single Proportion" and then "Edit Data" to enter the sample information. Hypotheses: \(H_{0}: p=0.5\) vs \(H_{a}: p<0.5\) Sample data: \(\hat{p}=38 / 100=0.38\) with \(n=100\)

Price and Marketing How influenced are consumers by price and marketing? If something costs more, do our expectations lead us to believe it is better? Because expectations play such a large role in reality, can a product that costs more (but is in reality identical) actually be more effective? Baba Shiv, a neuroeconomist at Stanford, conducted a study \(^{21}\) involving 204 undergraduates. In the study, all students consumed a popular energy drink which claims on its packaging to increase mental acuity. The students were then asked to solve a series of puzzles. The students were charged either regular price \((\$ 1.89)\) for the drink or a discount price \((\$ 0.89)\). The students receiving the discount price were told that they were able to buy the drink at a discount since the drinks had been purchased in bulk. The authors of the study describe the results: "the number of puzzles solved was lower in the reduced-price condition \((M=4.2)\) than in the regular-price condition \((M=5.8) \ldots p<0.0001 . "\) (a) What can you conclude from the study? How strong is the evidence for the conclusion? (b) These results have been replicated in many similar studies. As Jonah Lehrer tells us: "According to Shiv, a kind of placebo effect is at work. Since we expect cheaper goods to be less effective, they generally are less effective, even if they are identical to more expensive products. This is why brand-name aspirin works better than generic aspirin and why Coke tastes better than cheaper colas, even if most consumers can't tell the difference in blind taste tests." 22 Discuss the implications of this research in marketing and pricing.

Euchre One of the authors and some statistician friends have an ongoing series of Euchre games that will stop when one of the two teams is deemed to be statistically significantly better than the other team. Euchre is a card game and each game results in a win for one team and a loss for the other. Only two teams are competing in this series, which we'll call Team A and Team B. (a) Define the parameter(s) of interest. (b) What are the null and alternative hypotheses if the goal is to determine if either team is statistically significantly better than the other at winning Euchre? (c) What sample statistic(s) would they need to measure as the games go on? (d) Could the winner be determined after one or two games? Why or why not?

In Exercises 4.146 to \(4.149,\) hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample and give the significance level used. 4.148 Hypotheses: \(H_{0}: \rho=0\) vs \(H_{a}: \rho \neq 0 .\) In addition, in each case for which the results are significant, give the sign of the correlation. (a) \(95 \%\) confidence interval for \(\rho: \quad 0.07\) to 0.15 (b) \(90 \%\) confidence interval for \(\rho: \quad-0.39\) to -0.78 (c) \(99 \%\) confidence interval for \(\rho:-0.06\) to 0.03

Testing for a Gender Difference in Compassionate Rats In Exercise 3.80 on page 203 , we found a \(95 \%\) confidence interval for the difference in proportion of rats showing compassion, using the proportion of female rats minus the proportion of male rats, to be 0.104 to \(0.480 .\) In testing whether there is a difference in these two proportions: (a) What are the null and alternative hypotheses? (b) Using the confidence interval, what is the conclusion of the test? Include an indication of the significance level. (c) Based on this study would you say that female rats or male rats are more likely to show compassion (or are the results inconclusive)?
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