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Chapter 21: Problem 18

In the autumn, contests for the largest pumpkin are held in several counties in Ohio. Winners can weigh more that 1500 pounds. Two new varieties of pumpkin are to be compared to see which produces the largest pumpkins. To do this, 10 plots of land located in different regions of the state will be used. Half of each plot will be planted with one pumpkin variety and half with the other variety. You compare the weights of both varieties and use (a) the two-sample \(t\) test. (b) the matched pairs \(t\) test. (c) the one-sample \(t\) test.

Short Answer

Expert verified
The matched pairs \(t\) test (option b) should be used.

Step by step solution

01

Identify the type of data

In the experiment, for each plot of land, we have a pair of pumpkin weights: one from Variety A and one from Variety B. This implies a paired data structure for comparison.
02

Determine the purpose of the test

The experiment aims to compare two different varieties of pumpkins and see which one tends to produce larger pumpkins. Since both varieties are grown in the same environmental conditions within each plot, this comparison takes advantage of the paired nature of the data.
03

Choose the appropriate statistical test

Given that the data is paired, the appropriate test to use is the matched pairs \(t\) test. This test is specifically designed to handle paired samples, which are not independent but correlated.

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Key Concepts

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

Paired Data Structure
In experiments where two related samples are involved, we often use a paired data structure. This means that each data point in one sample has a corresponding data point in the other sample. In the context of pumpkin weights, each plot of land is divided into two sections, each growing a different variety of pumpkin.
This creates paired observations because both varieties experience similar conditions on the same plot. Such pairing helps in isolating and observing the effects of the different pumpkin species. The paired data structure is crucial for reducing variability caused by external factors, allowing for a clearer comparison of the varieties' performance.
Statistical Test Selection
Selecting the correct statistical test is a vital step in data analysis to ensure the results are valid. When deciding on a test, consider how the data is structured and what kind of inference you wish to make.
With paired data, as seen in the pumpkin experiment, the matched pairs t test is often the best choice because it accounts for the correlation between paired samples. If the samples were not paired, options like a two-sample t test might be more appropriate.
  • Matched Pairs T Test: Best for paired, dependent data.
  • Two-Sample T Test: Used for independent data comparing two groups.
Understanding these differences is key to effective data analysis.
Comparative Experiment Design
In comparative experiments, the goal is to compare two or more methods, treatments, or conditions. This pumpkin experiment is a classic example, where two pumpkin varieties are being compared.
The plots are designed to control environmental variables by using the same plot for both varieties. This design helps ensure that any differences in pumpkin size are due to the variety itself, not external factors. Key considerations in such designs include
  • Randomization: To prevent bias, treatments are randomly assigned.
  • Control for Confounding Variables: Ensure that extraneous variables do not skew results.
This structured approach helps to cleanly attribute differences to the variables under investigation.
Two-Sample T Test
The two-sample t test is used when comparing the means of two independent groups. It assumes that samples from both groups are collected independently and are normally distributed.
If our pumpkin varieties were planted on entirely separate plots with different conditions, this test might be appropriate. However, in a paired design, where each pair shares the same environment, the matched pairs t test is more effective.
In scenarios where you have independent samples, ensure your data meets assumptions like homogeneity of variance to apply the two-sample t test effectively.
One-Sample T Test
The one-sample t test is employed when comparing a sample mean to a known value or a population mean. Unlike other t tests, it does not compare between groups but measures against a standard.
For example, if we were testing whether pumpkins reach a certain average weight threshold, a one-sample t test would be suitable.
In our pumpkin scenario, the direct comparison between two varieties rather than a standard eliminates the need for this method.
Data Analysis in Statistics
Data analysis in statistics involves collecting, cleaning, transforming, and modeling data to discover useful information and support decision-making. In statistical analysis, choosing the right test like the matched pairs t test for paired data ensures accurate insights.
To successfully analyze data, follow these crucial steps:
  • Data Collection: Gather accurate and relevant data.
  • Data Cleaning: Correct or remove errors and inconsistencies.
  • Selection of Statistical Method: Determine the appropriate test based on data structure.
  • Interpretation: Understand the results within the context of your research.
Effective data analysis helps in drawing meaningful conclusions and making informed decisions.

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Most popular questions from this chapter

Do education programs for preschool children that follow the Montessori method perform better than other programs? A study compared five-year-old children in Milwaukee, Wisconsin, who had been enrolled in preschool programs from the age of three. \({ }^{15}\) (a) Explain why comparing children whose parents chose a Montessori school with children of other parents would not show whether Montessori schools perform better than other programs. (In fact, all the children in the study applied to the Montessori school. The school district assigned students to Montessori or other preschools by a random lottery.) (b) In all, 54 children were assigned to the Montessori school and 112 to other schools at age three. When the children were five, parents of 30 of the Montessori children and 25 of the others could be located and agreed to and subsequently participated in testing. This information reveals a possible source of bias in the comparison of outcomes. Explain why. (c) One of the many response variables was score on a test of ability to apply basic mathematics to solve problems. Here are summaries for the children who took this test: $$ \begin{array}{lccc} \hline \text { Group } & n & \mathrm{x}^{-\bar{x}} & \mathrm{~s} \\ \hline \text { Montessori } & 30 & 19 & 3.11 \\ \hline \text { Control } & 25 & 17 & 4.19 \\ \hline \end{array} $$ Is there evidence of a difference in the population mean scores? (The researchers used two-sided alternative hypotheses.)

Lamb's-quarter is a common weed that interferes with the growth of corn. An agriculture researcher planted corn at the same rate in 16 small plots of ground, then weeded the plots by hand to allow a fixed number of lamb's-quarter plants to grow in each meter of corn row. No other weeds were allowed to grow. Here are the yields of corn (bushels per acre) for only the experimental plots controlled to have one weed per meter of row and nine weeds per meter of row: \({ }^{9}\) $$ \begin{array}{l|llll} \hline \text { One weed/meter } & 166.2 & 157.3 & 166.7 & 161.1 \\ \hline \text { Nine weeds/meter } & 162.8 & 142.4 & 162.8 & 162.4 \\ \hline \end{array} $$ Explain carefully why a two-sample \(t\) confidence interval for the difference in mean yields may not be accurate.

The data you used in the previous two problems came from a random sample of students who took the SAT twice. The response rate was \(63 \%\), which is pretty good for nongovernment surveys, so let's accept that the respondents do represent all students who took the exam twice. Nonetheless, we can't be sure that coaching actually caused the coached students to gain more than the uncoached students. Explain briefly but clearly why this is so.

In a study of the effects of mood on evaluation of nutritious food, 208 subjects were randomly assigned to read either a happy story (to induce a positive mood) or a control (no story, neutral mood) group. Subjects were then asked to evaluate their attitude toward a certain indulgent food on a nine- point scale, with higher numbers indicating a more positive attitude toward the food. The following table summarizes data on the attitude rating: \({ }^{14}\) $$ \begin{array}{lccc} \hline \text { Group } & n & \mathrm{x}^{-} \bar{x} & \mathrm{~s} \\ \hline \text { Positive mood } & 104 & 4.30 & 2.05 \\ \hline \text { Neutral mood } & 104 & 5.50 & 1.74 \\ \hline \end{array} $$ (a) What are the standard errors of the sample means of the two groups? (b) What degrees of freedom does the conservative Option 2 use for twosample \(t\) procedures for these data? (c) Test the null hypothesis of no difference between the two group means against the two-sided alternative. Use the degrees of freedom from part (b).

To study whether higher predictions are rated as more accurate than lower predictions, 161 college students were presented a prediction by a basketball expert about the winner of an upcoming basketball game between teams A and B. All participants learned that the expert had carefully examined the two teams' history, players, and other information. Eighty students were randomly assigned to a group in which the expert claimed the chance that A would win was \(70 \%\). The other 81 students were assigned to a group in which the expert claimed that the chance B would win was \(30 \%\). All students rated the expert on his accuracy using a scale ranging from 0 to 20 , with higher scores indicating greater accuracy. Are the conditions for two-sample \(t\) inference satisfied? (a) Maybe: the SRS condition is OK but we need to look at the data to check Normality. (b) No: scores in a range between 0 and 20 can't be Normal. (c) Yes: the SRS condition is OK and large sample sizes make the Normality condition unnecessary.
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