Problem 52 - Basic Statistics for Business and Economics with Student

Question

52\. You need to estimate the mean number of travel days per year for pharmaceutical sales representatives. The mean of a small pilot study was 150 days, with a standard deviation of 14 days. If you must estimate the population mean within 2 days, how many sales representative in the pharmaceutical industry should you sample? Use the 90 percent confidence level.

Step-by-Step Solution

Answer

You need a sample size of 133 sales representatives.

Step 1: Understand the Problem

To estimate the mean number of travel days per year for pharmaceutical sales representatives, we need to find the required sample size for a given confidence level and margin of error.

Step 2: Identify Given Values

From the problem, we know the pilot study mean is 150 days, the standard deviation (\( \sigma \)) is 14 days, we want the estimate within 2 days, and we're using a 90% confidence level.

Step 3: Find the Z-Score for 90% Confidence Level

Look up the Z-score for a 90% confidence level. The Z-score splits 5% of the tail area on each side of the normal distribution. For 90%, the Z-score is approximately 1.645.

Step 4: Use the Margin of Error Formula

The formula for margin of error (E) is: \[ E = Z \times \frac{\sigma}{\sqrt{n}} \] Here, the margin of error is 2 days. Rearranging for \( n \) (sample size): \[ n = \left(\frac{Z \times \sigma}{E}\right)^2 \]

Step 5: Substitute Values into Formula

Substituting the known values into the rearranged formula: \[ n = \left(\frac{1.645 \times 14}{2}\right)^2 \] Calculate to find \( n \).

Step 6: Calculate the Sample Size

Perform the calculation: \[ n = \left(\frac{1.645 \times 14}{2}\right)^2 = \left(\frac{23.03}{2}\right)^2 = \left(11.515\right)^2 \approx 132.56 \] Round up to the nearest whole number because you can't survey a fraction of a person.

Step 7: Determine the Required Sample Size

The required sample size is 133 sales representatives to ensure the population mean is estimated within 2 days with 90% confidence.

Key Concepts

Confidence Level

When you are trying to estimate something, like the mean number of travel days for sales representatives, the confidence level tells you how sure you are that the true mean falls within your estimated range. With a 90% confidence level, it means that if you were to repeat this process multiple times, 90% of the calculated intervals would contain the true mean.
To find this, statisticians often use a Z-score, which is a value from the standard normal distribution that corresponds to the confidence level. For 90%, the Z-score is approximately 1.645.
A higher confidence level gives greater assurance but requires a larger sample size or allows for a greater margin of error.

Margin of Error

The margin of error is crucial in understanding how precise your estimate of the population mean is. In this context, it's the range above and below the sample mean in which you believe the true population mean will lie. In this exercise, a margin of error of 2 days means you expect the true mean to be no more than 2 days higher or lower than your estimate from the sample.
The formula for calculating the margin of error is: \[ E = Z \times \frac{\sigma}{\sqrt{n}} \] where:

• \( E \) is the margin of error.
• \( Z \) is the Z-score corresponding to your confidence level.
• \( \sigma \) is the standard deviation.
• \( n \) is the sample size.

A smaller margin of error means a more precise estimate, but it typically requires a larger sample size.

Standard Deviation

The standard deviation, often represented by \( \sigma \), measures how spread out the numbers in your data set are. In other words, it tells you how much the individual data points deviate from the mean. A small standard deviation indicates that the data points tend to be very close to the mean. A large standard deviation means the data points are spread out over a broader range of values.
In our example, a standard deviation of 14 days means that the travel days of sales representatives are spread out around the mean (150 days) by 14 days on average. This is used in the margin of error formula to account for variability in the data when calculating how many samples are needed to estimate the mean accurately.

Z-Score

A Z-Score is a statistical measurement that describes a value's relation to the mean of a group of values. Specifically, it indicates how many standard deviations an element is from the mean. In sample size calculation, it's used to determine the confidence interval.
For a 90% confidence level, the corresponding Z-score is 1.645. This means that 90% of the data falls within 1.645 standard deviations from the mean if the data follows a normal distribution.
To find the Z-score for your confidence level, you often refer to a Z-table, which provides the Z-scores for a range of probabilities, or use statistical software. This Z-score is then used in calculating the necessary sample size to ensure that statistical inferences drawn are within acceptable limits.