Problem 24 - Basic Statistics for Business & Economics

Question

A processor of carrots cuts the green top off each carrot, washes the carrots, and inserts six to a package. Twenty packages are inserted in a box for shipment. Each box of carrots should weigh 20.4 pounds. The processor knows that the standard deviation of box weight is 0.5 pound. The processor wants to know if the current packing process meets the 20.4 weight standard. How many boxes must the processor sample to be \(95 \%\) confident that the estimate of the population mean is within 0.2 pound?

Step-by-Step Solution

Answer

The processor needs to sample 25 boxes to be 95% confident.

Step 1: Identify the Known Values

We know that the standard deviation of the weight of the boxes is 0.5 pounds, the desired confidence level is 95%, and the margin of error is 0.2 pounds.

Step 2: Find the Z-value for the Desired Confidence Level

For a 95% confidence level, the Z-value is typically 1.96. This is derived from the standard normal distribution table.

Step 3: Apply the Sample Size Formula

Use the formula for sample size: \[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \]where \( \sigma \) is the standard deviation, \( E \) is the margin of error, and \( Z \) is the Z-value for the desired confidence level.

Step 4: Insert Known Values into the Formula

Substitute the known values into the formula: \[ n = \left( \frac{1.96 \cdot 0.5}{0.2} \right)^2 \]

Step 5: Calculate the Sample Size

Simplify the expression inside the parentheses first, \[ \frac{1.96 \cdot 0.5}{0.2} = 4.9 \] Then calculate the square of 4.9:\[ n = (4.9)^2 = 24.01 \]

Step 6: Round Up to the Nearest Whole Number

Since sample size must be a whole number, round up 24.01 to 25.

Key Concepts

Confidence Interval

A confidence interval gives a range of values within which we can be certain the true population parameter lies, to a point of confidence. For example, if we have a 95% confidence interval, that means that, statistically, if we repeated an experiment or sampled randomly multiple times, 95% of the intervals calculated from those experiments should contain the population parameter.

• The concept allows researchers to quantify the uncertainty in measurements or results.
• In practical terms, the confidence level tells you how sure you can be that the interval includes the parameter being estimated.

This particular problem requires the processor to understand the mean weight of the carrot boxes to ensure that they accurately meet the weight standard, which is ensured within an interval determined by the confidence interval technique.

Standard Deviation

Standard deviation measures how spread out the numbers in a data set are. In the context of this exercise, the standard deviation of box weight is given as \(0.5\) pounds. This figure indicates how much variation or "dispersion" exists from the average box weight.

• A smaller standard deviation means the values tend to be close to the mean.
• A larger deviation indicates that the values are more spread out.

In the sample size determination process, standard deviation plays a key role as it influences how widely data is expected to vary. Understanding how this metric impacts the calculation helps highlight potential issues with variability in box weight that the processor might need to address.

Margin of Error

The margin of error represents the extent of random sampling error in a survey’s results. In this case, the margin of error is \(0.2\) pounds, meaning that the processor wants to ensure that the estimate of the population mean won't deviate from the actual mean by more than \(0.2\) pounds.

• It reflects listeners' willingness for an acceptable range of difference?
• Helps gauge the precision of estimates – the smaller the margin, the more precise the estimate.

When calculating the sample size, the margin of error essentially provides the bounds within which the sample mean must fall concerning the true population mean. It's akin to setting the "wiggle room" allowed for when calculating how many samples are adequate to make meaningful conclusions.

Z-value

The Z-value, within the realm of statistics, is a measure that describes a value's relationship to the mean of a group of values. It signifies how many standard deviations an element is from the mean. For a 95% confidence level, the Z-value is typically \(1.96\). This number is crucial in deriving confidence intervals due to its tie to the standard normal distribution.

• This value is also known as the 'critical value' and varies with the confidence level.
• The chosen Z-value anchors the confidence interval width, directly affecting the interval's precision.

In the sample size formula, \( (Z \cdot \sigma / E)^2 \), the Z-value is utilized to adjust how expansive the interval is around the mean estimation. It ensures the desired confidence level is maintained without compromising the representativeness of the sample size.