91Ó°ÊÓ

What price do farmers get for their watermelon crops? In the third week of July, a random sample of 40 farming regions gave a sample mean of \(\bar{x}=\6.88\) per 100 pounds of watermelon. Assume that \(\sigma\) is known to be \(1.92\) per 100 pounds (Reference: Agricultural Statistics\(,\) U.S. Department of Agriculture). (a) Find a \(90 \%\) confidence interval for the population mean price (per 100 pounds) that farmers in this region get for their watermelon crop. What is the margin of error? (b) Sample Size Find the sample size necessary for a \(90 \%\) confidence level with maximal margin of error \(E=0.3\) for the mean price per 100 pounds of watermelon. (c) A farm brings 15 tons of watermelon to market. Find a \(90 \%\) confidence interval for the population mean cash value of this crop. What is the margin of error? Hint: 1 ton is 2000 pounds.

Step by step solution

01

Calculate the Z-Score for 90% Confidence Level

For a 90% confidence interval, the Z-score corresponds to 5% in each tail of the normal distribution. Therefore, the Z-score for a 90% confidence interval is approximately 1.645.

02

Calculate the Margin of Error (a)

Use the formula for margin of error: \[ E = Z \times \frac{\sigma}{\sqrt{n}} \] where \( Z = 1.645 \), \( \sigma = 1.92 \), and \( n = 40 \). Substitute the values to find the margin of error.

03

Apply the Confidence Interval Formula (a)

The confidence interval is \( \bar{x} \pm E \). Use the sample mean \( \bar{x} = 6.88 \) and the margin of error calculated in Step 2 to find the confidence interval for the mean price.

04

Solve for Sample Size (b)

Use the formula for sample size with maximum margin of error: \[ n = \left(\frac{Z \times \sigma}{E}\right)^2 \] Substitute \( Z = 1.645 \), \( \sigma = 1.92 \), and \( E = 0.3 \) to calculate the necessary sample size \( n \).

05

Convert Tons to Pounds and Prices (c)

15 tons is equivalent to \( 15 \times 2000 = 30000 \) pounds. Since the mean price is given per 100 pounds, convert this to a total of \( 300 \) units of measurement (x/100 pounds).

06

Calculate Total Mean Value and Margin of Error (c)

Calculate the total mean value for 300 units: \[ \text{Mean Value} = 300 \times 6.88 \] and the margin of error: \[ E = 300 \times \, \text{margin calculated in step 2} \].

07

Calculate Confidence Interval for Cash Value (c)

Multiply the confidence interval limits obtained in Step 3 by 300 to obtain the confidence interval for the cash value of the watermelon.

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Key Concepts

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

Margin of Error

The margin of error is a crucial component in statistics. It measures the amount of random sampling error in a survey's results. In simpler terms, it indicates how much the survey results could potentially differ from the actual population value. The **formula** used to calculate the margin of error is:
\[ E = Z \times \frac{\sigma}{\sqrt{n}} \]
Where:

• \( E \) is the margin of error
• \( Z \) is the Z-score, associated with the desired level of confidence
• \( \sigma \) is the population standard deviation
• \( n \) is the sample size

When you calculate the margin of error, you understand how much your sample mean might differ from the true population mean. By incorporating this into your confidence interval, you can express with more certainty how close your sample result is likely to be to the population parameter you're trying to estimate.

Sample Size Calculation

Determining the correct sample size is pivotal in any statistical study as it impacts the confidence level and accuracy of the results. To calculate the necessary sample size for a specific confidence level and margin of error, you can use the following formula:
\[ n = \left(\frac{Z \times \sigma}{E}\right)^2 \]
This formula ensures that the sample size you choose is sufficient to achieve your desired confidence level with a specified margin of error.

• For our scenario, with a 90% confidence level and a margin of error of 0.3, we substituted \( Z = 1.645 \), \( \sigma = 1.92 \), and \( E = 0.3 \) into the formula to compute the sample size.

Using this method helps to ensure that your findings accurately reflect the larger population.

Z-Score

A **Z-score** is a statistic that tells you where a particular data point lies in relation to the mean of a group of data. More specifically, it indicates how many standard deviations an element is from the mean.
A higher Z-score shows that a data point is further away from the mean compared to others. In the context of confidence intervals, the Z-score helps specify the level of certainty you wish to have.

• For a 90% confidence interval like in our exercise, a Z-score of approximately 1.645 is used.

This score reflects the probability that the true population mean lies within the calculated interval.

• Therefore, when we set the Z-score for a 90% confidence interval, we assert that there's a 90% chance our interval contains the actual population mean.

Population Mean

The **population mean** is a central value around which individual data points in a dataset cluster. Think of it as the average value of a characteristic across the entire population. In statistics, the population mean is denoted by the Greek letter \( \mu \).
To estimate this from a sample, we often calculate a point estimate, which is the sample mean, represented by \( \bar{x} \).

• In the watermelon example, the sample mean is \( \bar{x} = 6.88 \), and it's used to estimate the population mean.

The accuracy of the population mean estimation improves with the size of the sample and it becomes more representative of the entire population. Being able to accurately estimate the population mean helps in making informed business or research decisions.