Question

A random sampling of a company's monthly operating expenses for \(n=36\) months produced a sample mean of \(\$ 5474\) and a standard deviation of \(\$ 764\). Find a \(90 \%\) upper confidence bound for the company's mean monthly expenses.

Short answer

Answer: The 90% upper confidence bound for the company's mean monthly operating expenses is $5688.92.

Step-by-step solution

Identify the relevant statistics

First, we'll identify the relevant statistics from the problem: - Sample size (n) = 36 - Sample mean (x̄) = $5474 - Sample standard deviation (s) = $764

Determine the degrees of freedom and critical value

Since we are using the t-distribution, we'll need to find the degrees of freedom (df) and the critical value (t*) for a 90% one-sided confidence interval. - Degrees of freedom (df) = n - 1 = 36 - 1 = 35 - Confidence level = 90%, which leaves 10% in the upper tail (as it is one-sided). From the t-distribution table or using a calculator, we can find the critical value (t*) corresponding to 10% of the area in the upper tail for 35 degrees of freedom. The critical value t* = 1.689.

Calculate the standard error

Next, we need to calculate the standard error (SE) of the mean, which represents the uncertainty in estimating the population mean from our sample. The standard error is given by the formula: SE = s / √n = \(764 / √36 = \)764 / 6 = $127.33

Calculate the margin of error

Once we have the standard error, we can calculate the margin of error (ME) for our confidence interval. The margin of error is given by the formula: ME = t* × SE = 1.689 × \(127.33 = \)214.92

Calculate the upper confidence bound

Finally, we can calculate the upper confidence bound for our confidence interval. Since we are asked for the upper bound only, we calculate it by adding the margin of error to the sample mean: Upper confidence bound = x̄ + ME = \(5474 + \)214.92 = $5688.92

Interpret the results

We can now interpret the results of our calculations. We can be 90% confident that the company's mean monthly operating expenses are less than $5688.92. This means that there is only a 10% chance that the true mean monthly expenses are higher than this upper bound.

Key concepts

Sample Mean

The sample mean is a fundamental concept in statistics. It serves as an estimate of the true mean of the entire population from which a sample is drawn. In our exercise, the sample mean is given as $5474. This figure represents the average of the company's monthly operating expenses over 36 months.

To calculate the sample mean, you would add up all the data points in your sample and then divide by the number of data points. It gives us a point estimate of the population mean, providing a central value a typical member of the sample reflects.

• It is essential because it helps in making inferences about the population mean.
• It is sensitive to extreme values or outliers, which means it's affected by unusually high or low data points within the sample.

Understanding the sample mean helps us interpret statistical analyses and is crucial in constructing confidence intervals.

Standard Deviation

The standard deviation measures how much the values in a data set differ from the mean. It tells us how spread out the data points are. In our example, the standard deviation is \(764. This means that most of the company's monthly expenses are \)764 away from the average of $5474.

A high standard deviation indicates that the data points are spread out over a wide range of values, while a low standard deviation indicates that they are close to the mean. The formula to calculate the standard deviation is:

\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} \]

• It's crucial for determining the confidence interval because it quantifies the variability in the data sample.
• It is used to calculate the standard error, which is critical in making probabilistic statements about the population mean.

Overall, understanding the standard deviation helps us comprehend data variability and error margin in estimates.

T-Distribution

The t-distribution, often used when the sample size is small, allows for greater variability due to smaller sample sizes. It is particularly useful when the population standard deviation is unknown. Unlike the normal distribution, the t-distribution has heavier tails, meaning it is likely to produce values that fall far from the mean.

In our exercise, the degrees of freedom for the t-distribution are 35, which is calculated as the sample size minus one ( - 1). The degrees of freedom determine which t-distribution to use. For a 90% confidence interval, the critical value (t*) is 1.689. This value is used to calculate the margin of error.

• The t-distribution adjusts for sample sizes and is crucial for accurately estimating confidence intervals when dealing with a small sample size.
• As sample size increases, the t-distribution approaches the normal distribution.

Recognizing when to apply the t-distribution is key in statistics, especially when estimating population parameters from sample data.

Margin of Error

The margin of error represents the range of values below and above the sample mean in a confidence interval. It reflects the uncertainty or potential error in the sample estimate of the population mean. In our example, the margin of error is computed as approximately $214.92.

The formula to calculate it is:
\[ ME = t^* \times SE \]
where \(t^*\) is the critical value from the t-distribution and \(SE\) is the standard error. The margin of error gives us a boundary within which we expect the true population mean to lie.

• A larger margin of error suggests less confidence in the precision of the estimate.
• It is influenced by the variability of data and the sample size; greater sample sizes and less variability lead to smaller margins of error.

Understanding margin of error is vital as it quantifies uncertainty in statistical estimates, offering insights into the reliability of findings.