91Ó°ÊÓ

Popcorn Yvon Hopps ran an experiment to determine optimum power and time settings for microwave popcorn. His goal was to find a combination of power and time that would deliver high-quality popcorn with less than \(10 \%\) of the kernels left unpopped, on average. After experimenting with several bags, he determined that power 9 at 4 minutes was the best combination. To be sure that the method was successful, he popped 8 more bags of popcorn (selected at random) at this setting. All were of high quality, with the following percentages of uncooked popcorn: \(7,13.2,10,6,7.8,2.8,2.2,5.2 .\) Does the \(95 \%\) confidence interval suggest that he met his goal of an average of no more than \(10 \%\) uncooked kernels? Explain.

Step by step solution

01

Calculate Sample Mean and Sample Standard Deviation

First, calculate sample mean (average percentage of uncooked popcorn): add all the individual percentages of uncooked popcorn and divide by the total number of samples (8 in this case). For the sample standard deviation, subtract each percentage from the sample mean, square the result, sum up these squared values, divide by the number of samples minus 1, and finally, take the square root of this result.

02

Determine the Confidence Interval

Subsequently, use the formula for the confidence interval: \(\bar{x} ± Z * \frac{s}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(Z\) is the Z-score (for a \(95\%\) confidence level, the Z-score is approximately 1.96), \(s\) is the sample standard deviation, and \(n\) is the number of samples (8 in this case). This will give you the upper and lower bounds of the interval.

03

Interpret the Results

Lastly, interpret the results: if the confidence interval contains the value 10, it suggests that Yvon may not have met his goal since there is a chance -within the calculated confidence interval- that the average percentage of uncooked popcorn exceeds 10\%. If the interval does not contain the value 10, it suggests that Yvon has met his goal, as the average percentage of uncooked popcorn is less than 10\% with a \(95\%\) confidence level.

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Key Concepts

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

Sample Mean

When conducting experiments or analyzing data, the sample mean is a crucial concept. It represents the average value of a set of numbers and provides an estimate of the central tendency of the data. In Yvon's popcorn experiment, the sample mean helps determine the average percentage of uncooked kernels across the 8 bags of popcorn. To calculate the sample mean:

• Add together all the individual percentages of uncooked popcorn from each bag.
• Divide the total sum by the number of bags used in the experiment, which in this case is 8.

This result gives you the expected average percentage of uncooked popcorn, which can then be compared against the goal of less than 10% uncooked kernels.

Sample Standard Deviation

The sample standard deviation is another essential concept in statistical analysis. It measures the amount of variability or spread in a set of data points. For Yvon's experiment, the sample standard deviation tells us how much the percentages of uncooked kernels deviate from the average percentage. Here's how to compute it:

• Subtract the sample mean from each percentage of uncooked popcorn.
• Square each of the resulting differences.
• Sum all the squared values.
• Divide this total by the number of samples minus one, which in Yvon’s case is 7.
• Find the square root of the result from the previous step.

This value allows you to understand how consistent the uncooked percentage is across all bags sampled.

Z-score

The Z-score is a statistical measure that indicates the number of standard deviations a data point is from the mean. In the context of Yvon's experiment, the Z-score is used to calculate the confidence interval, helping us assess if the average percentage of uncooked kernels significantly deviates from the desired threshold. To find a Z-score:

• Identify the confidence level you’re working with, which is 95% in this case. For this confidence level, the Z-score is typically 1.96.

The Z-score is crucial in constructing a confidence interval for the sample mean, allowing you to understand the range in which the true population mean lies.

Statistical Analysis

Statistical analysis involves using mathematical methods to assess and interpret data. In Yvon’s experiment, statistical analysis helps us determine whether the chosen microwave settings are likely to produce popcorn with less than 10% of kernels unpopped.The key components involve:

• Calculating the sample mean and standard deviation to summarize the data.
• Constructing a confidence interval to estimate where the true population mean might lie. This is done using the formula \[ \bar{x} \pm Z * \frac{s}{\sqrt{n}} \]where \( \bar{x} \) is the sample mean, \( Z \) is the Z-score, \( s \) is the sample standard deviation, and \( n \) is the number of samples.
• Interpreting the results to see if they meet the goal. If the number 10 is within the confidence interval, Yvon might not have met his goal.

By applying these statistical techniques, Yvon can make informed decisions based on his data, ensuring the quality of microwave popcorn.