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Chapter 8: Problem 46

Exercises 8.46 to 8.52 refer to the data with analysis shown in the following computer output: \(\begin{array}{lrrrr}\text { Level } & \text { N } & \text { Mean } & \text { StDev } & \\ \text { A } & 6 & 86.833 & 5.231 & \\ \text { B } & 6 & 76.167 & 6.555 & \\ \text { C } & 6 & 80.000 & 9.230 & \\ \text { D } & 6 & 69.333 & 6.154 & \\ \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\ \text { Groups } & 3 & 962.8 & 320.9 & 6.64 & 0.003 \\ \text { Error } & 20 & 967.0 & 48.3 & & \\ \text { Total } & 23 & 1929.8 & & & \end{array}\) Is there evidence for a difference in the population means of the four groups? Justify your answer using specific value(s) from the output.

Short Answer

Expert verified
Yes, there is evidence for a difference in the population means of the four groups. This is because the P-value (0.003) is lower than the typical significance level (0.05), suggesting that at least one of the population means significantly differs from the others.

Step by step solution

01

Identify the P-value

From the provided table, identify the P-value given in the row labelled 'Groups'. This value is provided in the column labelled 'P' for the variable 'Groups'. In this case, the P-value is 0.003.
02

Compare the P-value to the significance level

The common significance level is 0.05. Compare the computed P-value (0.003) with this significance level. In this case, 0.003 < 0.05.
03

Decision Rule

Since the P-value is less than the significance level (伪=0.05), it suggests that at least one of the population means differs significantly from the rest. Hence, there is evidence to reject the null hypothesis. Therefore, there is evidence to conclude that the population means of the different groups are not all the same.

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Key Concepts

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

Population Means Comparison
Comparing population means is a fundamental aspect of ANOVA (Analysis of Variance), a statistical method used to analyze the differences between group means and their associated procedures.

In the context of the given exercise, the goal of ANOVA is to determine whether there are statistically significant differences among the mean scores of the four different groups (A, B, C, and D). The table from the exercise provides the means and the sample size for each group, along with the ANOVA summary output, which includes the degrees of freedom (DF), sum of squares (SS), mean squares (MS), F-statistic (F), and P-value.

The F-statistic is calculated by dividing the variance between the groups (MS of Groups) by the variance within the groups (MS of Error). The outcome of this calculation helps determine if the variance observed between group means is larger than what could be expected due to random chance.

If the calculated F-statistic is sufficiently large, it could indicate a significant difference in population means. However, to establish whether this difference is statistically significant, we must assess the P-value that corresponds with the observed F-statistic.
P-value Interpretation
The P-value is a critical concept in statistics used to interpret the results of hypothesis testing. It represents the probability of obtaining the observed results, or more extreme results, given that the null hypothesis is true.

With regard to our ANOVA analysis, the P-value is used to determine the evidence against the null hypothesis, which typically posits that there is no difference between group means. In simple terms, a smaller P-value suggests that there is stronger evidence to reject the null hypothesis.

In the provided exercise, a P-value of 0.003 is observed, which indicates a very low probability that the differences between the group means have occurred by chance alone. This P-value is crucial as it quantifies the evidence and helps decide whether the null hypothesis can be rejected. For the interpretation, the smaller the P-value, the greater the statistical significance of the observed differences.
Significance Level
The significance level, often denoted by \( \alpha \), is a threshold chosen by researchers to determine the point at which the results of a hypothesis test can be considered statistically significant. It is a pre-determined probability for erroneously rejecting the null hypothesis, known as a Type I error.

Commonly, a significance level of 0.05 (5%) is used in many scientific studies. If the P-value is less than or equal to the significance level, it suggests that the observed data are sufficiently inconsistent with the null hypothesis, and it is therefore rejected.

In the exercise, the P-value (0.003) is compared with the standard significance level (0.05). Since 0.003 is less than 0.05, this leads to the conclusion that there is statistically significant evidence against the null hypothesis. Therefore, we infer that the differences between the population means of the groups are unlikely to have arisen by random chance, signifying a meaningful difference in the means.

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Most popular questions from this chapter

Effects of Synchronization and Exertion on Closeness Exercise 8.16 on page 554 looks at possible differences in ratings of closeness to a group after doing a physical activity that involves either high or low levels of synchronization (HS or LS) and high or low levels of exertion (HE or LE). Students were randomly assigned to one of four groups with different combinations of these variables, and the change in their ratings of closeness to their group (on a 1 to 7 scale) were recorded. The data are stored in SynchronizedMovement and the means for each treatment group are given below, along with an ANOVA table that indicates a significant difference in the means at a \(5 \%\) level. \(\begin{array}{llrr}\text { Group } & \text { N } & \text { Mean } & \text { StDev } \\ \text { HS+HE } & 72 & 0.319 & 1.852 \\ \text { HS+LE } & 64 & 0.328 & 1.861 \\ \text { LS+HE } & 66 & 0.379 & 1.838 \\ \text { LS+LE } & 58 & -0.431 & 1.623\end{array}\) Analysis of Variance \(\begin{array}{lrrrrr}\text { Analysis or varlance } & & & \\ \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F-Value } & \text { P-Value } \\ \text { Group } & 3 & 27.04 & 9.012 & 2.77 & 0.042 \\ \text { Error } & 256 & 831.52 & 3.248 & & \\ \text { Total } & 259 & 858.55 & & & \end{array}\) The first three means look very similar, but the LS+LE group looks quite a bit different from the others. Is that a significant difference? Test this by comparing the mean difference in change in closeness ratings between the synchronized, high exertion activity group (HS+HE) and the nonsvnchronized. low exertion activity group (LS+LE).

Exercises 8.41 to 8.45 refer to the data with analysis shown in the following computer output: \(\begin{array}{lrrr}\text { Level } & \text { N } & \text { Mean } & \text { StDev } \\ \text { A } & 5 & 10.200 & 2.864 \\ \text { B } & 5 & 16.800 & 2.168 \\ \text { C } & 5 & 10.800 & 2.387\end{array}\) \(\begin{array}{lrrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\ \text { Groups } & 2 & 133.20 & 66.60 & 10.74 & 0.002 \\ \text { Error } & 12 & 74.40 & 6.20 & & \\ \text { Total } & 14 & 207.60 & & & \end{array}\) What is the pooled standard deviation? What degrees of freedom are used in doing inferences for these means and differences in means?

Data 4.1 introduces a study in mice showing that even low-level light at night can interfere with normal eating and sleeping cycles. In the full study, mice were randomly assigned to live in one of three light conditions: LD had a standard light/dark cycle, LL had bright light all the time, and DM had dim light when there normally would have been darkness. Exercises 8.28 to 8.34 in Section 8.1 show that the groups had significantly different weight gain and time of calorie consumption. In Exercises 8.55 and \(8.56,\) we revisit these significant differences. Body Mass Gain Computer output showing body mass gain (in grams) for the mice after four weeks in each of the three light conditions is shown, along with the relevant ANOVA output. Which light conditions give significantly different mean body mass gain? \(\begin{array}{lrrr}\text { Level } & \mathrm{N} & \text { Mean } & \text { StDev } \\ \text { DM } & 10 & 7.859 & 3.009 \\ \text { LD } & 9 & 5.987 & 1.786 \\ \text { LL } & 9 & 11.010 & 2.624\end{array}\) One-way ANOVA: BM4Gain versus Light \(\begin{array}{lrrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\\ \text { Light } & 2 & 116.18 & 58.09 & 8.96 & 0.001 \\ \text { Error } & 25 & 162.10 & 6.48 & & \\ \text { Total } & 27 & 278.28 & & & \end{array}\)

Drug Resistance and Dosing Exercise 8.39 on page 561 explores the topic of drug dosing and drug resistance by randomizing mice to four different drug treatment levels: untreated (no drug), light ( \(4 \mathrm{mg} / \mathrm{kg}\) for 1 day), moderate \((8 \mathrm{mg} / \mathrm{kg}\) for 1 day), or aggressive ( \(8 \mathrm{mg} / \mathrm{kg}\) for 5 or 7 days). Exercise 8.39 found that, contrary to conventional wisdom, higher doses can actually promote drug resistance, rather than prevent it. Here, we further tease apart two different aspects of drug dosing: duration (how many days the drug is given for) and amount per day. Recall that four different response variables were measured; two measuring drug resistance (density of resistant parasites and number of days infectious with resistant parasites) and two measuring health (body mass and red blood cell density). In Exercise 8.39 we don't find any significant differences in the health responses (Weight and \(R B C)\) so we concentrate on the drug resistance measures (ResistanceDensity and DaysInfectious) in this exercise. The data are available in DrugResistance and we are not including the untreated group. (a) Investigate duration by comparing the moderate treatment with the aggressive treatment (both of which gave the same amount of drug per day, but for differing number of days). Which of the two resistance response variables (ResistanceDensity and DaysInfectious) have means significantly different between these two treatment groups? For significant differences, indicate which group has the higher mean. (b) Investigate amount per day by comparing the light treatment with the moderate treatment (both of which lasted only 1 day, but at differing amounts). Which of the two resistance response variables have means significantly different between these two treatment groups? For significant differences, indicate which group has the higher mean. (c) Does duration or amount seem to be more influential (at least within the context of this study)? Why?

Exercises 8.41 to 8.45 refer to the data with analysis shown in the following computer output: \(\begin{array}{lrrr}\text { Level } & \text { N } & \text { Mean } & \text { StDev } \\ \text { A } & 5 & 10.200 & 2.864 \\ \text { B } & 5 & 16.800 & 2.168 \\ \text { C } & 5 & 10.800 & 2.387\end{array}\) \(\begin{array}{lrrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\ \text { Groups } & 2 & 133.20 & 66.60 & 10.74 & 0.002 \\ \text { Error } & 12 & 74.40 & 6.20 & & \\ \text { Total } & 14 & 207.60 & & & \end{array}\) Find a \(90 \%\) confidence interval for the difference in the means of populations \(\mathrm{B}\) and \(\mathrm{C}\).
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