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Chapter 4: Problem 86

State the conclusion of the test based on this p-value in terms of "Reject \(H_{0} "\) or "Do not reject \(H_{0} "\), if we use a \(5 \%\) significance level. p-value \(=0.2531\)

Short Answer

Expert verified
As the p-value (0.2531) is greater than the significance level (0.05), we Do not reject the null hypothesis \(H_{0}\).

Step by step solution

01

Understand the Significance Level

In this case, the significance level is \(5\% = 0.05\). It is the probability of rejecting the null hypothesis \(H_{0}\), when it is true. It is also the maximum probability one - if doing the experiment over and over - would risk to reject the null hypothesis \(H_{0}\), when it is true.
02

Know the Given p-value

The given p-value in this exercise is 0.2531. If we recall, p-value is the minimum level of significance at which we can reject the null hypothesis \(H_{0}\). So, a higher p-value indicates that we should not reject the null hypothesis.
03

Comparison

Now we compare the given p-value (0.2531) with the significance level (0.05). We see that the p-value (0.2531) is higher than our significance level (0.05).

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Key Concepts

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

Significance Level
The significance level, often denoted as alpha (\( \alpha \)), is a critical threshold in hypothesis testing. Its primary role is to help us decide if the evidence we gather is strong enough to reject the null hypothesis (\( H_0 \)). It's usually set before the data is collected and analyzed. Common choices for significance levels include 0.05 (\( 5\% \)), 0.01, or 0.10, depending on the field of study and the level of confidence you need in your results.
  • The \( 5\% \) level implies you're willing to accept a \( 5\% \) risk of concluding that a difference exists when there isn鈥檛 one.
  • This level ensures that the Type I error, which occurs when the null hypothesis is true but rejected, is minimized to \( 5\% \)
We use significance levels to maintain consistency and rigor in statistical testing, helping researchers draw reliable conclusions.
p-value
The p-value is a fundamental concept in hypothesis testing, representing the probability of obtaining results at least as extreme as the observed ones, given that the null hypothesis (\( H_0 \)) is true. Essentially, the p-value tells us how likely our observed data would occur under the assumption of the null hypothesis. A small p-value indicates that the observed data is unlikely under \( H_0 \), thus leading us to consider rejecting it.
  • If our p-value is less than the significance level, we reject \( H_0 \).
  • A p-value greater than the significance level suggests insufficient evidence to reject \( H_0 \)
For the problem at hand, the given p-value is 0.2531, which is higher than the typical threshold of 0.05. This implies that there's enough probability that the null hypothesis could still be true, leading us to not reject \( H_0 \).
Null Hypothesis
The null hypothesis (\( H_0 \)) serves as a starting point or baseline in hypothesis testing. It generally posits that there is no effect, no difference, or no change. For example, in a clinical trial, the null hypothesis might state that a new drug has no impact compared to a placebo.
  • The goal of hypothesis testing is to determine whether there is enough statistical evidence to reject \( H_0 \), making the alternative hypothesis the more likely explanation.
  • To make this decision, we use the significance level and the p-value as tools to measure the strength of the evidence.
It is important for \( H_0 \) to be a precise statement to allow for accurate statistical measurement. In our exercise, since the p-value is 0.2531, which is greater than the significance level of 0.05, we conclude there isn't enough evidence to reject \( H_0 \).This suggests the null hypothesis is consistent with the observed data.

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Most popular questions from this chapter

Using the definition of a p-value, explain why the area in the tail of a randomization distribution is used to compute a p-value.

4.150 Approval from the FDA for Antidepressants The FDA (US Food and Drug Administration) is responsible for approving all new drugs sold in the US. In order to approve a new drug for use as an antidepressant, the FDA requires two results from randomized double-blind experiments showing the drug is more effective than a placebo at a \(5 \%\) level. The FDA does not put a limit on the number of times a drug company can try such experiments. Explain, using the problem of multiple tests, why the FDA might want to rethink its guidelines. 4.151 Does Massage Really Help Reduce Inflammation in Muscles? In Exercise 4.112 on page \(301,\) we learn that massage helps reduce levels of the inflammatory cytokine interleukin-6 in muscles when muscle tissue is tested 2.5 hours after massage. The results were significant at the \(5 \%\) level. However, the authors of the study actually performed 42 different tests: They tested for significance with 21 different compounds in muscles and at two different times (right after the massage and 2.5 hours after). (a) Given this new information, should we have less confidence in the one result described in the earlier exercise? Why? (b) Sixteen of the tests done by the authors involved measuring the effects of massage on muscle metabolites. None of these tests were significant. Do you think massage affects muscle metabolites? (c) Eight of the tests done by the authors (including the one described in the earlier exercise) involved measuring the effects of massage on inflammation in the muscle. Four of these tests were significant. Do you think it is safe to conclude that massage really does reduce inflammation?

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 \(^{25}\) 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 \((M=5.8) \ldots p<.0001 . "\) (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."26 Discuss the implications of this research in marketing and pricing.

Giving a Coke/Pepsi taste test to random people in New York City to determine if there is evidence for the claim that Pepsi is preferred.

Data 4.3 on page 265 introduces a situation in which a restaurant chain is measuring the levels of arsenic in chicken from its suppliers. The question is whether there is evidence that the mean level of arsenic is greater than \(80 \mathrm{ppb},\) so we are testing \(H_{0}: \mu=80\) vs \(H_{a}:\) \(\mu>80,\) where \(\mu\) represents the average level of arsenic in all chicken from a certain supplier. It takes money and time to test for arsenic, so samples are often small. Suppose \(n=6\) chickens from one supplier are tested, and the levels of arsenic (in ppb) are: \(\begin{array}{llllll}68, & 75, & 81, & 93, & 95, & 134\end{array}\) (a) What is the sample mean for the data? (b) Translate the original sample data by the appropriate amount to create a new dataset in which the null hypothesis is true. How do the sample size and standard deviation of this new dataset compare to the sample size and standard deviation of the original dataset? (c) Write the six new data values from part (b) on six cards. Sample from these cards with replacement to generate one randomization sample. (Select a card at random, record the value, put it back, select another at random, until you have a sample of size \(6,\) to match the original sample size.) List the values in the sample and give the sample mean. (d) Generate 9 more simulated samples, for a total of 10 samples for a randomization distribution. Give the sample mean in each case and create a small dotplot. Use an arrow to locate the original sample mean on your dotplot.
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