Question

Suppose that \(10 \%\) of the fields in a given agricultural area are infested with the sweet potato whitefly. One hundred fields in this area are randomly selected and checked for whitefly. a. What is the average number of fields sampled that are infested with whitefly? b. Within what limits would you expect to find the number of infested fields, with probability approximately \(95 \% ?\) c. What might you conclude if you found that \(x=25\) fields were infested? Is it possible that one of the characteristics of a binomial experiment is not satisfied in this experiment? Explain.

Short answer

Answer: The average number of infested fields is 10. The range within which we can find infested fields with a 95% probability is 4 to 16. If 25 fields are found infested, it is beyond the upper limit of our 95% probability range, suggesting possible issues with the experiment, such as sampling issues or variations in-field conditions. Further investigation would be required to determine the cause of this deviation.

Step-by-step solution

Calculate the average number of infested fields

To find the average number of infested fields, we need to multiply the total number of sampled fields (100) by the probability of finding an infested field (10%). So, Average number of infested fields = \(100 * 0.10 = 10\) Therefore, the average number of infested fields is 10.

Find the values within a 95% probability range

To find the range from which we can find infested fields within a 95% probability, we need to calculate the standard deviation and apply it to the mean (average number of infested fields). Standard deviation: \(\sigma = \sqrt{n * p * (1 - p)}\), where \(n = 100\) (number of samples) and \(p = 0.10\) (probability of infestation). Standard deviation: \(\sigma = \sqrt{100 * 0.10 * (1 - 0.10)} = \sqrt{9} = 3\) Now, let's find the range boundaries by applying a 95% probability, which corresponds to \(\pm 1.96 * 3\) standard deviations: Lower limit: \(10 - 1.96 * 3 = 10 - 5.88 \approx 4\) Upper limit: \(10 + 1.96 * 3 = 10 + 5.88 \approx 16\) So, the number of infested fields will be within the range of 4 to 16 with a probability of 95%.

Evaluate the situation when 25 fields are infested

If we find that 25 fields are infested, we can see that it is way above the upper limit (16) of our 95% probability range. In this case, there is a possibility that one of the binomial characteristics is not satisfied in this experiment. This could be due to sampling issues, variations in-field conditions leading to variations in infestation probabilities, or other factors. Further investigation would be required to determine the exact reason for this deviation.

Key concepts

Probability

Probability is a fundamental concept in statistics that measures the likelihood of a specific event occurring. In our exercise involving fields infested with sweet potato whitefly, probability helps us determine different outcomes, such as finding infested fields.

Here, the probability (\(p\)) of a single field being infested is stated as 10%, or 0.10. This value is crucial in calculating expected results and standard deviation, aiding in predictions about real-world scenarios.

• To find the average number of infested fields, multiply the total fields checked by the probability of infestation: \(Average = n * p = 100 * 0.10 = 10\).
• This result implies, on average, 10 out of 100 fields will be infested, a useful baseline for further statistical analysis.

Probability allows for understanding and quantifying uncertainty, deeply rooted in making informed decisions based on statistical data.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It tells us how spread out the numbers are around the mean (average).
In the exercise, we used standard deviation to understand the variability of the number of infested fields from the average number. The formula applied is \(\sigma = \sqrt{n * p * (1 - p)}\), with:

• \(n = 100\): the total number of fields sampled.
• \(p = 0.10\): probability of infestation.

Plugging these into the formula gives us:
\(\sigma = \sqrt{100 * 0.10 * (1 - 0.10)} = \sqrt{9} = 3\).

This standard deviation tells us that the number of infested fields typically deviate by 3 from the average of 10. A smaller standard deviation would indicate the data points tend to be close to the mean, whereas a larger deviation shows they are more spread out.
Standard deviation is critical in understanding the data variance and providing insights into how consistent the outcome might be.

Confidence Interval

The confidence interval gives a range of values within which we expect our estimated parameter to lie, with a certain degree of confidence. In our context, this is crucial in inferring about the number of infested fields.
For the exercise, a 95% confidence interval was calculated using the mean and standard deviation. Here, the 95% confidence interval means we are 95% confident that the true mean number of infested fields falls within this range.

• The mean (\(\mu\)) is 10.
• The standard deviation (\(\sigma\)) is 3.

Using the z-score for 95% confidence (\(\pm 1.96\)), the interval boundaries are:

• Lower limit: \( 10 - 1.96 * 3 \approx 4\).
• Upper limit: \( 10 + 1.96 * 3 \approx 16\).

This suggests that the number of infested fields should typically be between 4 and 16 for most samples. When results, like 25 infested fields, fall outside this range, it signals possible issues or the need for further examination. Understanding confidence intervals helps in assessing the reliability of statistical estimates and the precision of these estimates in practical scenarios.