Problem 38 - Basic Statistics for Business and Economics with Student

Question

The Human Relations Department of Electronics, Inc. would like to include a dental plan as part of the benefits package. The question is: How much does a typical employee and his or her family spend per year on dental expenses? A sample of 45 employees reveals the mean amount spent last year was \(\$ 1,820,\) with a standard deviation of \(\$ 660 .\) a. Construct a 95 percent confidence interval for the population mean. b. The information from part (a) was given to the president of Electronics, Inc. He indicated he could afford \(\$ 1,700\) of dental expenses per employee. Is it possible that the population mean could be \(\$ 1,700 ?\) Justify your answer.

Step-by-Step Solution

Answer

Yes, the population mean could be $1700, as it falls within the confidence interval (1625.45, 2014.55).

Step 1: Identify the given information

We are given a sample size ( 45 ) of employees, a sample mean ( 1820 ), and a sample standard deviation ( 660 ). We need to construct a 95% confidence interval for the population mean dental expenses.

Step 2: Determine the confidence interval formula

We use the formula for the confidence interval for the population mean: \( \bar{x} \pm t \times \left(\frac{s}{\sqrt{n}}\right) \), where \( \bar{x} \) is the sample mean, \( s \) is the sample standard deviation, \( n \) is the sample size, and \( t \) is the t-value for the desired confidence level.

Step 3: Find the t-value for a 95% confidence level

Since the sample size is 45, we'll use 44 degrees of freedom (df = n - 1). For a 95% confidence level, look up the t-value in a t-distribution table or use a calculator to find it. The t-value is approximately 2.015.

Step 4: Calculate the standard error of the mean

The standard error (SE) is calculated as \( \frac{s}{\sqrt{n}} = \frac{660}{\sqrt{45}} \). Calculate this value to use in the confidence interval formula.

Step 5: Compute the confidence interval

Plug the values into the formula from Step 2: \( 1820 \pm 2.015 \times \frac{660}{\sqrt{45}} \). Calculate the margin of error and then find the confidence interval by adding and subtracting the margin of error from the sample mean.

Step 6: Interpret the confidence interval

After calculating, the 95% confidence interval ends up being approximately (1625.45, 2014.55). This means we are 95% confident that the true mean of dental expenses per employee falls within this range.

Step 7: Evaluate if the population mean could be $1700

Since \\(1700 falls within the 95% confidence interval of (1625.45, 2014.55), it is possible that the population mean could be \\)1700. This means the company could potentially afford the dental expenses for employees.

Key Concepts

Sample Mean

The sample mean is crucial in understanding how a group of observations behaves. In our exercise, this refers to the average dental expenses calculated from the sample of employees. For the 45 employees surveyed, the sample mean was \( 1,820 \). This figure helps us estimate the population mean, which represents the average dental expense for all employees, not just those in the sample. When interpreting a sample mean:

• It acts as an unbiased estimator of the population mean if the sample is random and representative.
• A higher sample size generally provides a more accurate estimate of the population mean.
• Consider the spread of data around this mean to understand variability.

The reliability of a sample mean as an estimate of the population mean increases with the sample size.

Standard Deviation

The standard deviation is a key measure in statistics that indicates how spread out the values in a data set are. In our case, the standard deviation of \( 660 \) provides insight into the variation of dental expenses among employees. Understanding standard deviation helps us grasp the amount of variability and diversity in the observed expenses.Key concepts related to standard deviation:

• A smaller standard deviation means the data points are closer to the mean, indicating less variability.
• A larger standard deviation suggests a wider spread of data points, showing more variability.
• It is used in calculating the standard error, which is essential when constructing confidence intervals.

Remember, in a normal distribution, most of the data will typically fall within one or two standard deviations of the mean.

T-Distribution

The t-distribution is an essential concept when constructing confidence intervals, especially when dealing with smaller sample sizes. It is similar to a normal distribution but has heavier tails, which accounts for the added uncertainty with small samples. In constructing the 95% confidence interval for the dental expenses, we used the t-distribution because our sample size was relatively small (45 employees). Understanding the t-distribution:

• It is used instead of the normal distribution when the population standard deviation is unknown and the sample size is small.
• The shape of the t-distribution becomes closer to the normal distribution as the sample size increases.
• The degrees of freedom, which are one less than the sample size, play a critical role in determining the t-value.

This distribution helps us estimate the margin of error and ultimately, the confidence interval.

Population Mean

The population mean is the average of all possible observations in the population, and it is what we ultimately aim to estimate through our sample mean. In our scenario, it answers the question: What is the average dental expense for all employees?Using a confidence interval, we infer that the actual population mean is likely to fall within a specified range, with a certain level of confidence. Here, we calculated a 95% confidence interval for the population mean and found that it spans from \( 1,625.45 \) to \( 2,014.55 \). Important points about the population mean:

• It is a key parameter in inferential statistics, helping make broad conclusions about a population based on sample data.
• Though the exact value is often unknown, methods like confidence intervals allow us to estimate it with some degree of certainty.
• The confidence interval's range includes plausible values for the population mean based on our sample data.

Understanding and calculating the population mean is critical in decision-making, especially in assessing affordability for company benefits.