Problem 1 - Basic Statistics for Business and Economics with Student

Question

A sample of 49 observations is taken from a normal population with a standard deviation of \(10 .\) The sample mean is \(55 .\) Determine the 99 percent confidence interval for the population mean.

Step-by-Step Solution

Answer

The 99% confidence interval is approximately 51.318 to 58.682.

Step 1: Identify the Given Information

We know that the sample size is 49, the standard deviation of the population is 10, the sample mean is 55, and we're seeking a 99% confidence interval. We will use these values in our calculations.

Step 2: Determine the Z-score for a 99% Confidence Level

For a 99% confidence interval, you need the Z-score that corresponds to the middle 99% of the standard normal distribution. This is typically found in a Z-table or using statistical software. The Z-score for a 99% confidence interval is approximately 2.576.

Step 3: Calculate the Standard Error of the Mean

The standard error of the mean (SEM) is calculated by dividing the standard deviation by the square root of the sample size. In this case: \[SEM = \frac{10}{\sqrt{49}} = \frac{10}{7} = 1.4286.\]

Step 4: Calculate the Margin of Error

The margin of error (ME) is found by multiplying the Z-score by the standard error of the mean. So:\[ME = 2.576 \times 1.4286 \approx 3.682.\]

Step 5: Compute the Confidence Interval

To find the confidence interval, add and subtract the margin of error from the sample mean:\[\text{Lower limit} = 55 - 3.682 = 51.318,\]\[\text{Upper limit} = 55 + 3.682 = 58.682.\]

Step 6: State the Confidence Interval

The 99% confidence interval for the population mean is from approximately 51.318 to 58.682.

Key Concepts

Sample Size

Understanding sample size is crucial in statistics. It tells us how many observations are included in a sample. In other words, it's the number of data points collected or chosen from the entire population for analysis. In this exercise, the sample size is 49.

The sample size is an essential factor because it impacts the precision of our estimates. A larger sample size generally leads to more reliable and accurate results. This is because larger samples are likely to represent the population better.

• Helps improve the accuracy of statistical estimates.
• Affects the margin of error in confidence intervals.
• Contributes to the power of a statistical test.

By examining 49 observations here, we trust that the sample encapsulates the relevant features of the entire population, allowing us to make inferences about the population mean with a specified level of confidence.

Standard Deviation

Standard deviation is a measure that shows the amount of variation or dispersion in a set of values. It tells us how much the values in a data set deviate from the mean of the sample. In this particular example, the standard deviation of the population is given as 10.

A lower standard deviation indicates that the data points are close to the mean, while a higher standard deviation shows that they are spread out over a large range of values.

• Reflects data consistency.
• Influences the confidence interval width by impacting the standard error.
• Helps interpret data spread and variability.

In our scenario, knowing the standard deviation allows us to calculate the standard error, which in turn helps us determine more about the confidence interval for our estimates.

Standard Error of the Mean

The standard error of the mean (SEM) indicates how far the sample mean of the data is likely to be from the true population mean. It is calculated by dividing the standard deviation by the square root of the sample size, as shown in the steps.

For this exercise: \[SEM = \frac{10}{\sqrt{49}} = 1.4286.\] The SEM provides a glimpse into the reliability of the sample mean. A smaller SEM implies a more accurate estimate of the population mean.

• Determined by the standard deviation and sample size.
• Critical for calculating confidence intervals.
• Indicates the precision of the sample mean estimate.

By understanding SEM, one can interpret how well the sample mean represents the population mean and how much one should trust the mean result using the available data.

Z-score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations. For confidence intervals, a Z-score helps determine the number of standard deviations a point is from the mean.

In the exercise, we obtained a Z-score of approximately 2.576 for a 99% confidence level. This means that the interval we are defining encompasses 99% of the normal distribution.

• Derived from standard normal distribution tables or software.
• Assists in calculating confidence interval bounds.
• Varies depending on the desired confidence level.

Overall, Z-scores are fundamental in constructing confidence intervals, ensuring that we correctly assess the level of certainty for statistical estimates such as the population mean.