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Chapter 5: Problem 32

Assume that hybridization experiments are conducted with peas having the property that for offspring, there is a 0.75 probability that a pea has green pods (as in one of Mendel's famous experiments). Assume that offspring peas are randomly selected in groups of 16. a. Find the mean and standard deviation for the numbers of peas with green pods in the groups of 16. b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high. c. Is a result of 7 peas with green pods a result that is significantly low? Why or why not?

Short Answer

Expert verified
Mean: 12, Std Dev: 1.732. Thresholds: Low - 8.536, High - 15.464. Yes, 7 peas is significantly low.

Step by step solution

01

Understand the problem

We are given a probability of 0.75 that a pea has green pods. We need to find the mean and standard deviation of peas with green pods in a group of 16, use the range rule of thumb to find significantly low or high results, and determine if 7 peas with green pods is significantly low.
02

Find the mean

The mean for a binomial distribution is given by the formula \(\mu = np\) where \(n\) is the number of trials and \(p\) is the probability of success. Here, \(n = 16\) and \(p = 0.75\). \[ \mu = 16 \times 0.75 = 12 \]
03

Find the standard deviation

The standard deviation for a binomial distribution is given by \[ \sigma = \sqrt{np(1-p)} \] Using \(n = 16\) and \(p = 0.75\), \[ \sigma = \sqrt{16 \times 0.75 \times 0.25} = \sqrt{3} \approx 1.732 \]
04

Use the range rule of thumb

The range rule of thumb states that values significantly low or high can be found as \(\mu - 2\sigma\) and \(\mu + 2\sigma\). Using \(\mu = 12\) and \(\sigma = 1.732\), \[ \text{Low threshold} = 12 - 2 \times 1.732 \approx 8.536 \] \[ \text{High threshold} = 12 + 2 \times 1.732 \approx 15.464 \]
05

Determine if 7 peas is significantly low

Comparing 7 peas with our low threshold of 8.536, we see that 7 is less than 8.536. Thus, 7 peas with green pods is significantly low.

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Key Concepts

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

Mean Calculation
When dealing with binomial distributions, the mean tells us the expected number of successes. For example, in the problem provided, we need to calculate how many peas out of 16 are expected to have green pods if each pea has a 0.75 probability of having a green pod. This is calculated using the formula \(\mu = np\), where \(n\) represents the number of trials (16 peas) and \(p\) is the probability of success (0.75). Plugging in these values, we get \[ \mu = 16 \times 0.75 = 12 \] This mean value suggests that, on average, we can expect 12 peas to have green pods out of a group of 16.
Standard Deviation Calculation
The next step is to determine the standard deviation, which gives us an idea of how much variation there is from the mean. In a binomial distribution, the standard deviation is found using the formula \(\sigma = \sqrt{np(1-p)}\). Here, \(n = 16\) and \(p = 0.75\). After calculating, we see: \[ \sigma = \sqrt{16 \times 0.75 \times 0.25} = \sqrt{3} \approx 1.732 \] This value indicates that the number of peas with green pods will typically vary by about 1.732 from the mean of 12.
Range Rule of Thumb
The range rule of thumb is a simple way to determine what counts as significantly low or high values in a data set. You can find the significantly low and high values by using the mean \(\mu\) and the standard deviation \(\sigma\). The thresholds are given by: \[ \text{Low threshold} = \mu - 2\sigma \approx 12 - 2 \times 1.732 \approx 8.536 \] \[ \text{High threshold} = \mu + 2\sigma \approx 12 + 2 \times 1.732 \approx 15.464 \] Values below 8.536 are considered significantly low, while those above 15.464 are significantly high.
Significantly Low Values
To determine if a certain result is significantly low, compare it to the low threshold found using the range rule of thumb. Given our calculated low threshold of approximately 8.536, we see that a result of 7 peas with green pods is less than 8.536. Hence, the result of 7 peas with green pods is significantly low. This evaluation helps us identify if a certain outcome is unusual or unexpected compared to typical values predicted by our binomial distribution.

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Most popular questions from this chapter

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