/*! 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 22 The following data refers to yie... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 22

The following data refers to yield of tomatoes ( \(\mathrm{kg} / \mathrm{plot}\) ) for four different levels of salinity; salinity level here refers to electrical conductivity \((\mathrm{EC})\), where the chosen levels were \(\mathrm{EC}=1.6,3.8,6.0\), and \(10.2 \mathrm{nmhos} / \mathrm{cm}:\) $$ \begin{array}{rrrrrr} 1.6 & 59.5 & 53.3 & 56.8 & 63.1 & 58.7 \\ 3.8 & 55.2 & 59.1 & 52.8 & 54.5 & \\ 6.0 & 51.7 & 48.8 & 53.9 & 49.0 & \\ 10.2 & 44.6 & 48.5 & 41.0 & 47.3 & 46.1 \end{array} $$ Use the \(F\) test at level \(\alpha=.05\) to test for any differences in true average yield due to the different salinity levels.

Short Answer

Expert verified
Perform ANOVA and compare the F-statistic to the critical value. Reject \(H_0\) if \(F\) exceeds the critical value.

Step by step solution

01

State the Hypotheses

We need to test if there are significant differences in the average yield of tomatoes due to different salinity levels.- Null Hypothesis \( H_0 \): There is no difference in the true average yields among the different salinity levels.- Alternative Hypothesis \( H_a \): There is a difference in the true average yields among the different salinity levels.
02

Analyze the Experiment Design

Identify that this is an ANOVA (Analysis of Variance) problem as we are comparing means from more than two groups. There are four different groups based on salinity levels, each with replicates.
03

Calculate Group Means

Find the mean yield for each salinity level:- \(\mathrm{EC} = 1.6\): Mean = \(\frac{59.5 + 53.3 + 56.8 + 63.1 + 58.7}{5} = 58.28\)- \(\mathrm{EC} = 3.8\): Mean = \(\frac{55.2 + 59.1 + 52.8 + 54.5}{4} = 55.40\)- \(\mathrm{EC} = 6.0\): Mean = \(\frac{51.7 + 48.8 + 53.9 + 49.0}{4} = 50.85\)- \(\mathrm{EC} = 10.2\): Mean = \(\frac{44.6 + 48.5 + 41.0 + 47.3 + 46.1}{5} = 45.50\)
04

Calculate Overall Mean

Compute the overall mean of all observations:Overall mean = \(\frac{59.5 + 53.3 + 56.8 + 63.1 + 58.7 + 55.2 + 59.1 + 52.8 + 54.5 + 51.7 + 48.8 + 53.9 + 49.0 + 44.6 + 48.5 + 41.0 + 47.3 + 46.1}{18} = 52.77\)
05

Calculate Sum of Squares

Calculate the sum of squares required for ANOVA: - Total Sum of Squares (SST): Measure the total variability in the data. - Between Group Sum of Squares (SSB): Measure the variability between the group means. - Within Group Sum of Squares (SSW): Measure the variability within each group.
06

Obtain Mean Sum of Squares

Calculate the Mean Squares:- Mean Square Between (MSB) = \(\frac{SSB}{k-1}\), where \(k\) is the number of groups.- Mean Square Within (MSW) = \(\frac{SSW}{N-k}\), where \(N\) is the total number of observations.
07

Compute the F-statistic

The F-statistic is given by:\[ F = \frac{MSB}{MSW} \]
08

Determine the Critical Value and Make Decision

Determine the critical value from the F-distribution table using \( k-1 \) and \( N-k \) degrees of freedom. Compare the computed F-statistic with the critical value:- If \( F \) > critical value, reject \( H_0 \).- If \( F \) 鈮� critical value, fail to reject \( H_0 \).

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Key Concepts

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

F-test
The F-test is a crucial technique in statistics, especially when performing Analysis of Variance (ANOVA). It is used to determine if there are significant differences between the means of three or more groups. In the context of the exercise, the F-test helps to compare the average yields of tomatoes across different salinity levels. By conducting an F-test, we can assess whether the differences in yields are too large to be attributed to random sampling error alone.
This is accomplished by examining the variance between the group means in relation to the variance within each group.

The F-statistic is calculated as follows:
  • First, calculate the Mean Square Between (MSB), which represents the variability due to the differences between the group means.
  • Then, calculate the Mean Square Within (MSW), which reflects the variability within each group.
  • Finally, divide the MSB by the MSW to obtain the F-statistic.
If this F-statistic is significantly larger than what would be expected due to chance, we may conclude that there are meaningful differences between the group means.
Hypothesis Testing
Hypothesis Testing is a fundamental statistical method used to make decisions based on data. In the given exercise, hypothesis testing allows us to assess whether salinity levels significantly affect tomato yields. At the heart of hypothesis testing is the formulation of two contrasting statements: the null hypothesis (\( H_0 \) ) and the alternative hypothesis (\( H_a \) ).
  • The null hypothesis (\( H_0 \) ) posits that there is no difference in the average yields across the different salinity levels. It assumes that any variation observed is due to random chance.
  • The alternative hypothesis (\( H_a \) ), on the other hand, contends that there is indeed a significant difference in yields attributable to the varying salinity levels.
Adopting a significance level of \( \alpha = 0.05 \) , we use the F-test to evaluate these hypotheses. If the test statistic falls beyond a critical threshold as determined by the F-distribution table, the null hypothesis is rejected. This indicates that the differences in mean yields are statistically significant.
Sum of Squares
Sum of Squares (SS) is a measure of variability in a dataset and is a fundamental component of ANOVA. It quantifies the deviation of individual data points from the mean, providing insight into the overall spread or dispersion of the data. In the context of the exercise, three types of sum of squares are essential.
  • Total Sum of Squares (SST): This represents the total variance observed in the data. It accounts for the variability of all tomato yield observations from the overall mean.
  • Between Group Sum of Squares (SSB): This measures the variance attributed to the differences between the group means. It is crucial for assessing the variability due to different salinity levels.
  • Within Group Sum of Squares (SSW): This looks at the variance within each salinity group, effectively measuring the individual variability within each group.
Understanding these components is key to determining whether the observed differences in group means are statistically significant as part of the F-test.
Mean Square Between
The Mean Square Between (MSB) is a pivotal value when performing ANOVA. It is derived by dividing the Between Group Sum of Squares (SSB) by the degrees of freedom associated with the groups, which is calculated as \( k - 1 \), where \( k \) is the number of groups.

In the context of the exercise, the MSB provides a measure of how much the group means deviate from the overall mean. It reflects the consistency of the effects of different salinity levels on tomato yields.

  • A higher MSB indicates greater variability between the group means compared to the overall mean, suggesting a potential significant effect of salinity levels.
  • MSB helps to determine the F-statistic by providing a numerator that, when divided by the Mean Square Within (MSW), shows whether this between-group variability is more than expected under the null hypothesis.
By studying the MSB, researchers can better understand the role of external factors鈥攊n this scenario, salinity levels鈥攐n the variability observed in the data.

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Most popular questions from this chapter

Numerous factors contribute to the smooth running of an electric motor ("Increasing Market Share Through Improved Product and Process Design: An Experimental Approach," Quality Engineering, 1991: 361-369). In particular, it is desirable to keep motor noise and vibration to a minimum. To study the effect that the brand of bearing has on motor vibration, five different motor bearing brands were examined by installing each type of bearing on different random samples of six motors. The amount of motor vibration (measured in microns) was recorded when each of the 30 motors was running. The data for this study follows. State and test the relevant hypotheses at significance level \(.05\), and then carry out a multiple comparisons analysis if appropriate. $$ \begin{array}{llllllll} \text { Brand 1 } & 13.1 & 15.0 & 14.0 & 14.4 & 14.0 & 11.6 & 13.68 \\ \text { Brand 2 } & 16.3 & 15.7 & 17.2 & 14.9 & 14.4 & 17.2 & 15.95 \\ \text { Brand 3 } & 13.7 & 13.9 & 12.4 & 13.8 & 14.9 & 13.3 & 13.67 \\ \text { Brand 4 } & 15.7 & 13.7 & 14.4 & 16.0 & 13.9 & 14.7 & 14.73 \\ \text { Brand 5 } & 13.5 & 13.4 & 13.2 & 12.7 & 13.4 & 12.3 & 13.08 \end{array} $$

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