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Chapter 6: Problem 18

A hybridization experiment begins with four peas having yellow pods and one pea having a green pod. Two of the peas are randomly selected with replacement from this population. a. After identifying the 25 different possible samples, find the proportion of peas with yellow pods in each of them, then construct a table to describe the sampling distribution of the proportions of peas with yellow pods. b. Find the mean of the sampling distribution. c. Is the mean of the sampling distribution [from part (b)] equal to the population proportion of peas with yellow pods? Does the mean of the sampling distribution of proportions always equal the population proportion?

Short Answer

Expert verified
The mean of the sampling distribution is 0.8, equal to the population proportion. The mean of the sampling distribution of proportions always equals the population proportion.

Step by step solution

01

List Possible Samples

Create a list of all possible samples. Since peas are selected with replacement, there are 5 options (4 yellow, 1 green) for each of the 2 selections. This gives us a total of 25 samples: YY, YG, GY, GG, etc.
02

Find Proportion of Yellow Pods in Each Sample

For each of the 25 samples, calculate the proportion of yellow pods. For example, in sample 'YY', the proportion is 1 (100% yellow). In sample 'YG', the proportion is 0.5 (50% yellow).
03

Construct the Sampling Distribution Table

Construct a table listing each unique proportion of yellow pods and how often each proportion appears among the 25 samples.
04

Calculate the Mean of the Sampling Distribution

Calculate the mean of the proportions found in Step 3 by summing the proportions and dividing by the number of samples.
05

Compare with Population Proportion

Compare the mean of the sampling distribution to the population proportion (4 yellow out of 5 peas, or 0.8). Analyze if they are equal, and discuss whether the mean of the sampling distribution of proportions is always equal to the population proportion.

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Key Concepts

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

proportion of yellow pods
In a given set of samples, the proportion of yellow pods can be found by examining each sample and calculating the ratio of yellow pods to the total number of pods. For example, if our sample is 'YY' (where both peas are yellow), the proportion of yellow pods is 1 (or 100%). If the sample is 'YG' (one yellow, one green), the proportion is 0.5 (or 50%). By doing this for each sample, we create a detailed understanding of how often yellow pods appear in different combinations.
mean of sampling distribution
The mean of the sampling distribution is a central concept in statistics. It represents the average of all the sampling proportions. To calculate this, you sum up all the proportions and divide by the total number of samples. If we have 25 samples and the proportions for yellow pods are provided for each sample, we add all these proportions together and then divide by 25. This gives us an average proportion, providing insight into the expected value of yellow pods across repeated sampling.
population proportion
The population proportion is an overall measure that tells us the fraction of yellow pods present in the entire population. In our example, there are 4 yellow pods out of 5 total peas, giving us a population proportion of 4/5 or 0.8 (80%). This value is crucial as it serves as a benchmark against which we compare our sample proportions. By analyzing how closely our sample means align with this population proportion, we can understand the nature of our sampling process.
sampling distribution table
A sampling distribution table helps to organize the different sample proportions and indicates how often each proportion occurs. For example, if we have proportions like 1, 0.5, and 0 appearing several times among the 25 samples, we can tabulate this data to showcase the distribution. This table will list the unique proportions and their frequencies, allowing us to visualize and analyze the distribution patterns. Each row in the table might look something like this:
  • Proportion: 1, Frequency: 4
  • Proportion: 0.5, Frequency: 12
  • Proportion: 0, Frequency: 9
This visualization aids in understanding the variability and central tendency in the sampling proportions.

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Most popular questions from this chapter

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Use the data in the table below for sitting adult males and females (based on anthropometric survey data from Gordon, Churchill, et al.). These data are used often in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats. (Hint: Draw a graph in each case.) $$ \begin{aligned} &\text { Sitting Back-to-Knee Length (inches) }\\\ &\begin{array}{l|c|c|c} \hline & \text { Mean } & \text { St. Dev. } & \text { Distribution } \\ \hline \text { Males } & 23.5 \text { in. } & 1.1 \text { in. } & \text { Normal } \\ \hline \text { Females } & 22.7 \text { in. } & 1.0 \text { in. } & \text { Normal } \\ \hline \end{array} \end{aligned} $$ Find the probability that a male has a back-to-knee length between \(22.0\) in. and \(24.0\) in.
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