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Chapter 4: Problem 146

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: \(H_{0}: \mu=15\) vs \(H_{a}: \mu \neq 15\) (a) 95\% confidence interval for \(\mu: \quad 13.9\) to 16.2 (b) 95\% confidence interval for \(\mu: \quad 12.7\) to 14.8 (c) 90\% confidence interval for \(\mu: \quad 13.5\) to 16.5

Short Answer

Expert verified
Based on the analysis, for the first and third cases, the confidence intervals include the hypothesized mean of 15. Therefore, we do not reject the null hypothesis. However, in the second case, the confidence interval does not include the hypothesized mean, so we reject the null hypothesis and accept the alternative hypothesis at a 95% significance level.

Step by step solution

01

Analyzing the 95% Confidence Interval for the first case

In this case, the 95% confidence interval for mu is from 13.9 to 16.2. This interval does cover our hypothesized mean of 15. Therefore, we do not have enough evidence to reject the null hypothesis. There is a 95% confidence level.
02

Analyzing the 95% Confidence Interval for the second case

In this case, the confidence interval for mu is from 12.7 to 14.8. This interval does not include our hypothesized mean of 15. Therefore, we have enough evidence at the 95% confidence level to reject the null hypothesis and accept the alternative hypothesis.
03

Analyzing the 90% Confidence Interval for the third case

In this case, the 90% confidence interval for mu is from 13.5 to 16.5. This interval includes our hypothesized mean of 15. Therefore, we do not have enough evidence to reject the null hypothesis. There is a 90% confidence level.

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Key Concepts

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

Null Hypothesis
The null hypothesis, often denoted as \( H_0 \), is an initial statement made about a population parameter. It's put forward as a proposition that indicates no effect or no difference, essentially a statement of status quo. In our exercise, the null hypothesis is \( H_0: \mu = 15 \). This means that we are hypothesizing the population mean \( \mu \) is 15.The null hypothesis acts as a default or starting point. When conducting a statistical test, we seek evidence to decide whether this hypothesis can be rejected or not. This decision is typically made based on sample data and statistical tests, where we compute a statistic and compare it to a theoretical distribution to understand how surprising our data is, assuming the null hypothesis is true.If the sample data show that the statistic is significantly different from what the null hypothesis predicts, it may give us reason to reject \( H_0 \). However, if the data do not strongly contradict \( H_0 \), we would not reject it, maintaining the assumption of no effect.
Alternative Hypothesis
The alternative hypothesis, represented as \( H_a \) or \( H_1 \), stands in contrast to the null hypothesis. It's what a researcher aims to provide evidence for through their data analysis. In our given problem, the alternative hypothesis is \( H_a: \mu eq 15 \). This suggests a belief that the population mean \( \mu \) is not equal to 15. The role of the alternative hypothesis is crucial when we analyze data. It indicates that there is a genuine effect, change, or difference that contradicts the null hypothesis. The burden of proof lies in showing that the data collected are incompatible with \( H_0 \), and thus, \( H_a \) is a more plausible explanation.The alternative hypothesis can be two-sided or one-sided:
  • Two-sided: Suggests the parameter is not equal to a certain value, as in our exercise \( H_a : \mu eq 15 \).
  • One-sided: Suggests the parameter is either greater than or less than a certain value.
By considering the alternative hypothesis, we explore other possibilities beyond the null hypothesis in our investigation.
Statistical Significance
Statistical significance is a fundamental concept determining whether the result of a test is due to chance or represents a true effect in the population. It is often measured with \( p \)-value, the probability of observing the data, or something more extreme, assuming the null hypothesis is true.In our exercise involving confidence intervals, we interpret statistical significance in terms of whether or not the computed interval includes the value specified in the null hypothesis. If a confidence interval does not contain the null hypothesis value, we typically conclude there is statistical significance at a certain level (e.g., 95%).For example:
  • If \( H_0: \mu = 15 \) and a 95% confidence interval is from 12.7 to 14.8, we consider the result statistically significant because 15 is not within this interval.
  • Conversely, if the interval runs from 13.9 to 16.2, 15 is included, signifying lack of statistical significance at the 95% level.
The level of confidence chosen (90%, 95%, etc.) reflects how frequently the interval would capture the true mean in repeated samples. Thus, a statistically significant result indicates that the effect found is unlikely to be a random outcome, but rather, likely indicative of the true state of affairs in the population.

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Most popular questions from this chapter

Car Window Skin Cancer? A new study suggests that exposure to UV rays through the car window may increase the risk of skin cancer. \(^{43}\) The study reviewed the records of all 1050 skin cancer patients referred to the St. Louis University Cancer Center in 2004 . Of the 42 patients with melanoma, the cancer occurred on the left side of the body in 31 patients and on the right side in the other 11 . (a) Is this an experiment or an observational study? (b) Of the patients with melanoma, what proportion had the cancer on the left side? (c) A bootstrap \(95 \%\) confidence interval for the proportion of melanomas occurring on the left is 0.579 to \(0.861 .\) Clearly interpret the confidence interval in the context of the problem. (d) Suppose the question of interest is whether melanomas are more likely to occur on the left side than on the right. State the null and alternative hypotheses. (e) Is this a one-tailed or two-tailed test? (f) Use the confidence interval given in part (c) to predict the results of the hypothesis test in part (d). Explain your reasoning. (g) A randomization distribution gives the p-value as 0.003 for testing the hypotheses given in part (d). What is the conclusion of the test in the context of this study? (h) The authors hypothesize that skin cancers are more prevalent on the left because of the sunlight coming in through car windows. (Windows protect against UVB rays but not UVA rays.) Do the data in this study support a conclusion that more melanomas occur on the left side because of increased exposure to sunlight on that side for drivers?

Flaxseed and Omega-3 Studies have shown that omega-3 fatty acids have a wide variety of health benefits. Omega- 3 oils can be found in foods such as fish, walnuts, and flaxseed. A company selling milled flaxseed advertises that one tablespoon of the product contains, on average, at least \(3800 \mathrm{mg}\) of ALNA, the primary omega-3. (a) The company plans to conduct a test to ensure that there is sufficient evidence that its claim is correct. To be safe, the company wants to make sure that evidence shows the average is higher than \(3800 \mathrm{mg}\). What are the null and alternative hypotheses? (b) Suppose, instead, that a consumer organization plans to conduct a test to see if there is evidence against the claim that the product contains an average of \(3800 \mathrm{mg}\) per tablespoon. The consumer organization will only take action if it finds evidence that the claim made by the company is false and the actual average amount of omega- 3 is less than \(3800 \mathrm{mg}\). What are the null and alternative hypotheses?

Income East and West of the Mississippi For a random sample of households in the US, we record annual household income, whether the location is east or west of the Mississippi River, and number of children. We are interested in determining whether there is a difference in average household income between those east of the Mississippi and those west of the Mississippi. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) What statistic(s) from the sample would we use to estimate the difference? (b) What statistic(s) from the sample would we use to estimate the difference?

In Exercises 4.107 to \(4.111,\) null and alternative hypotheses for a test are given. Give the notation \((\bar{x},\) for example) for a sample statistic we might record for each simulated sample to create the randomization distribution. $$ H_{0}: \mu=15 \text { vs } H_{a}: \mu<15 $$

Red Wine and Weight Loss Resveratrol, an ingredient in red wine and grapes, has been shown to promote weight loss in rodents. A recent study \(^{19}\) investigates whether the same phenomenon holds true in primates. The grey mouse lemur, a primate, demonstrates seasonal spontaneous obesity in preparation for winter, doubling its body mass. A sample of six lemurs had their resting metabolic rate, body mass gain, food intake, and locomotor activity measured for one week prior to resveratrol supplementation (to serve as a baseline) and then the four indicators were measured again after treatment with a resveratrol supplement for four weeks. Some p-values for tests comparing the mean differences in these variables (before vs after treatment) are given below. In parts (a) to (d), state the conclusion of the test using a \(5 \%\) significance level, and interpret the conclusion in context. (a) In a test to see if mean resting metabolic rate is higher after treatment, \(p=0.013\). (b) In a test to see if mean body mass gain is lower after treatment, \(p=0.007\) (c) In a test to see if mean food intake is affected by the treatment, \(p=0.035\). (d) In a test to see if mean locomotor activity is affected by the treatment, \(p=0.980\) (e) In which test is the strongest evidence found? The weakest? (f) How do your answers to parts (a) to (d) change if the researchers make their conclusions using a stricter \(1 \%\) significance level? (g) For each p-value, give an informal conclusion in the context of the problem describing the level of evidence for the result. (h) The sample only included six lemurs. Do you think that we can generalize to the population of all lemurs that body mass gain is lower on average after four weeks of a resveratrol supplement? Why or why not?
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