/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 104 Figure 4.25 shows a scatterplot ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 104

Figure 4.25 shows a scatterplot of the acidity (pH) for a sample of \(n=53\) Florida lakes vs the average mercury level (ppm) found in fish taken from each lake. The full dataset is introduced in Data 2.4 on page 71 and is available in FloridaLakes. There appears to be a negative trend in the scatterplot, and we wish to test whether there is significant evidence of a negative association between \(\mathrm{pH}\) and mercury levels. (a) What are the null and alternative hypotheses? (b) For these data, a statistical software package produces the following output: $$ r=-0.575 \quad p \text { -value }=0.000017 $$ Use the p-value to give the conclusion of the test. Include an assessment of the strength of the evidence and state your result in terms of rejecting or failing to reject \(H_{0}\) and in terms of \(\mathrm{pH}\) and mercury. (c) Is this convincing evidence that low \(\mathrm{pH}\) causes the average mercury level in fish to increase? Why or why not?

Short Answer

Expert verified
The null hypothesis which posits no relationship between pH and mercury levels is rejected due to a very small p-value. This suggests a negative relationship between pH level and mercury level in the fish taken from the Florida lakes. However, this does not confirm that low pH is the cause of the increased average mercury level.

Step by step solution

01

Define the null and alternative hypotheses.

The null hypothesis \(H_0\) is that there is no association between pH and mercury level. On the other hand, the alternative hypothesis \(H_1\) is that there is a negative association between pH and mercury level. In mathematical terms, for \(H_0\), the correlation is \(0\) and for \(H_1\), the correlation is less than \(0\).
02

Interpret the p-value.

The p-value is \(0.000017\), it's exceedingly small, and typically, if the p-value is less than \(0.05\), this indicates strong evidence against the null hypothesis, and we reject the null hypothesis.
03

Formulate a conclusion based on the p-value.

Based on the very small p-value, there is strong evidence against the null hypothesis. Therefore, \(H_0\) is rejected in favor of \(H_1\), suggesting there is indeed a negative association between pH and mercury level.
04

Discuss causation.

Although a significant correlation (negative in this case) is observed, it cannot be concluded that low pH causes an increase in average mercury level. This is because correlation does not imply causation. Statistical results only provide evidence of an association, and do not establish a causative relationship, which would require a different type of study design.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Scatterplot Analysis
Scatterplot analysis is a visual representation used in statistics to examine the relationship between two quantitative variables. In our scenario, this involves pH levels in Florida lakes and the associated mercury levels in fish. Each point on the scatterplot depicts a pair of values, demonstrating how these two variables might be connected.

A scatterplot is beneficial because it quickly reveals trends and patterns, such as positive or negative correlations, clusters, or outliers. In this exercise, the scatterplot presented shows a negative trend, meaning as the pH decreases, mercury levels tend to increase.

To interpret this plot effectively, we should ask questions such as:
  • Are most of the points close to a trend line, indicating a strong relationship?
  • Does the plot exhibit any significant outliers that do not fit the pattern?
  • Is the trend upward or downward, pointing to a positive or negative correlation?
Such insights drive hypotheses and statistical testing to further explore these relationships.
Correlation vs Causation
Understanding the distinction between correlation and causation is crucial in data analysis. Correlation indicates a statistical relationship or association between two variables. However, just because two variables change together doesn't mean one causes the other to change.

In the given exercise, the correlation coefficient ( ") was -0.575, suggesting a moderate negative correlation between pH level and mercury in fish. This implies that as pH decreases, mercury content generally increases. However, this does not infer causation.

Some important points to consider about correlation vs causation:
  • Correlation: Quantified by the correlation coefficient (r), ranging from -1 to 1. Negative values indicate inverse relationships, while positive ones point to direct relationships.
  • Causation: Requires more than just correlation, often through controlled experiments that rule out outside variables.
  • Confounding Variables: Other factors can cause both variables to shift, giving the impression of causality.
Remember, observing a correlation is just the first step. Establishing causation demands thorough investigation and careful experimental design.
P-Value Interpretation
The p-value in hypothesis testing measures the strength of evidence against the null hypothesis. It's the probability of obtaining test results at least as extreme as the observed data, assuming the null hypothesis is true.

In our exercise, the p-value was reported as 0.000017, which is very small. This significantly low value suggests strong evidence against the null hypothesis. Generally, if the p-value is less than the significance level (often set at 0.05), we reject the null hypothesis.

Key points about p-value interpretation:
  • When the p-value is low, it indicates our observed data is unlikely under the null hypothesis, leading to its rejection.
  • If the p-value exceeds 0.05, we do not have sufficient evidence to reject the null hypothesis.
  • The p-value does not tell us about the size or importance of the effect, just the significance of the results.
For our scenario, the low p-value reinforced the conclusion that there is indeed a negative association between pH levels and mercury in fish, supporting the alternative hypothesis.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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.

Data 4.3 on page 265 discusses a test to determine if the mean level of arsenic in chicken meat is above 80 ppb. If a restaurant chain finds significant evidence that the mean arsenic level is above \(80,\) the chain will stop using that supplier of chicken meat. The hypotheses are $$ \begin{array}{ll} H_{0}: & \mu=80 \\ H_{a}: & \mu>80 \end{array} $$ where \(\mu\) represents the mean arsenic level in all chicken meat from that supplier. Samples from two different suppliers are analyzed, and the resulting p-values are given: Sample from Supplier A: p-value is 0.0003 Sample from Supplier B: p-value is 0.3500 (a) Interpret each p-value in terms of the probability of the results happening by random chance. (b) Which p-value shows stronger evidence for the alternative hypothesis? What does this mean in terms of arsenic and chickens? (c) Which supplier, \(A\) or \(B\), should the chain get chickens from in order to avoid too high a level of arsenic?

Translating Information to Other Significance Levels Suppose in a two-tailed test of \(H_{0}: \rho=0\) vs \(H_{a}: \rho \neq 0,\) we reject \(H_{0}\) when using a \(5 \%\) significance level. Which of the conclusions below (if any) would also definitely be valid for the same data? Explain your reasoning in each case. (a) Reject \(H_{0}: \rho=0\) in favor of \(H_{a}: \rho \neq 0\) at a \(1 \%\) significance level. (b) Reject \(H_{0}: \rho=0\) in favor of \(H_{a}: \rho \neq 0\) at a \(10 \%\) significance level. (c) Reject \(H_{0}: \rho=0\) in favor of the one-tail alternative, \(H_{a}: \rho>0,\) at a \(5 \%\) significance level, assuming the sample correlation is positive.

Mating Choice and Offspring Fitness: MiniExperiments Exercise 4.153 explores the question of whether mate choice improves offspring fitness in fruit flies, and describes two seemingly identical experiments yielding conflicting results (one significant, one insignificant). In fact, the second source was actually a series of three different experiments, and each full experiment was comprised of 50 different mini-experiments (runs), 10 each on five different days. (a) Suppose each of the 50 mini-experiments from the first study were analyzed individually. If mating choice has no impact on offspring fitness, about how many of these \(50 \mathrm{p}\) -values would you expect to yield significant results at \(\alpha=0.05 ?\) (b) The 50 p-values, testing the alternative \(H_{a}\) : \(p_{C}>p_{N C}\) (proportion of flies surviving is higher in the mate choice group) are given below: $$ \begin{array}{lllllllllll} \text { Day 1: } & 0.96 & 0.85 & 0.14 & 0.54 & 0.76 & 0.98 & 0.33 & 0.84 & 0.21 & 0.89 \\ \text { Day 2: } & 0.89 & 0.66 & 0.67 & 0.88 & 1.00 & 0.01 & 1.00 & 0.77 & 0.95 & 0.27 \\ \text { Day 3: } & 0.58 & 0.11 & 0.02 & 0.00 & 0.62 & 0.01 & 0.79 & 0.08 & 0.96 & 0.00 \\ \text { Day 4: } & 0.89 & 0.13 & 0.34 & 0.18 & 0.11 & 0.66 & 0.01 & 0.31 & 0.69 & 0.19 \\ \text { Day 5: } & 0.42 & 0.06 & 0.31 & 0.24 & 0.24 & 0.16 & 0.17 & 0.03 & 0.02 & 0.11 \end{array} $$ How many are actually significant using \(\alpha=0.05 ?\) (c) You may notice that two p-values (the fourth and last run on day 3 ) are 0.00 when rounded to two decimal places. The second of these is actually 0.0001 if we report more decimal places. This is very significant! Would it be appropriate and/or ethical to just report this one run, yielding highly statistically significant evidence that mate choice improves offspring fitness? Explain. (d) You may also notice that two of the p-values on day 2 are 1 (rounded to two decimal places). If we had been testing the opposite alternative, \(H_{a}:\) \(p_{C}

The consumption of caffeine to benefit alertness is a common activity practiced by \(90 \%\) of adults in North America. Often caffeine is used in order to replace the need for sleep. One study \(^{24}\) compares students' ability to recall memorized information after either the consumption of caffeine or a brief sleep. A random sample of 35 adults (between the ages of 18 and 39 ) were randomly divided into three groups and verbally given a list of 24 words to memorize. During a break, one of the groups takes a nap for an hour and a half, another group is kept awake and then given a caffeine pill an hour prior to testing, and the third group is given a placebo. The response variable of interest is the number of words participants are able to recall following the break. The summary statistics for the three groups are in Table 4.9. We are interested in testing whether there is evidence of a difference in average recall ability between any two of the treatments. Thus we have three possible tests between different pairs of groups: Sleep vs Caffeine, Sleep vs Placebo, and Caffeine vs Placebo. (a) In the test comparing the sleep group to the caffeine group, the p-value is \(0.003 .\) What is the conclusion of the test? In the sample, which group had better recall ability? According to the test results, do you think sleep is really better than caffeine for recall ability? (b) In the test comparing the sleep group to the placebo group, the p-value is 0.06 . What is the conclusion of the test using a \(5 \%\) significance level? If we use a \(10 \%\) significance level? How strong is the evidence of a difference in mean recall ability between these two treatments? (c) In the test comparing the caffeine group to the placebo group, the p-value is 0.22 . What is the conclusion of the test? In the sample, which group had better recall ability? According to the test results, would we be justified in concluding that caffeine impairs recall ability? (d) According to this study, what should you do before an exam that asks you to recall information?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.