Question

A simple random sample of 60 items resulted in a sample mean of \(80 .\) The population standard deviation is \(\sigma=15\) a. Compute the \(95 \%\) confidence interval for the population mean. b. Assume that the same sample mean was obtained from a sample of 120 items. Provide a \(95 \%\) confidence interval for the population mean. c. What is the effect of a larger sample size on the interval estimate?

Short answer

The 95% confidence interval for the population mean with a sample size of 60 items is approximately [76.18, 83.82]. When using a sample size of 120 items, the 95% confidence interval is approximately [77.31, 82.69]. A larger sample size results in a more precise estimate of the population mean, as demonstrated by the narrower confidence interval with the larger sample size.

Step-by-step solution

Find the Z-score for a 95% confidence level

To find the Z-score for a 95% confidence level, we need to look for the Z-value that corresponds to the area of 0.975 in the standard normal distribution table, since a 95% confidence level corresponds to the middle 95% of the distribution. From the table, we find that the Z-score is 1.96. **Step 2: Calculate the confidence interval for a sample size of 60**

Calculate confidence interval for n = 60

Given that the sample mean is 80, the population standard deviation is 15, and the sample size is 60, we can plug these values into our confidence interval formula: Confidence Interval = \(80 \pm (1.96 * \frac{15}{\sqrt{60}})\) Compute the margin of error: \(1.96 * \frac{15}{\sqrt{60}} \approx 3.82\) And hence, the confidence interval is: \(80 \pm 3.82 \approx [76.18, 83.82]\) **Step 3: Calculate the confidence interval for a sample size of 120**

Calculate confidence interval for n = 120

Now, we will calculate the confidence interval for a sample size of 120, with the same sample mean and population standard deviation: Confidence Interval = \(80 \pm (1.96 * \frac{15}{\sqrt{120}})\) Compute the margin of error: \(1.96 * \frac{15}{\sqrt{120}} \approx 2.69\) And hence, the confidence interval is: \(80 \pm 2.69 \approx [77.31, 82.69]\) **Step 4: Discuss the effect of a larger sample size on the interval estimate**

Compare the two confidence intervals

Comparing the confidence intervals for the two sample sizes (60 and 120), we can see that as the sample size increases, the width of the confidence interval decreases. This means that a larger sample size results in a more precise estimate of the population mean.

Key concepts

Sample Mean

The sample mean, often represented with the symbol \( \bar{x} \), plays a crucial role in statistics as it serves as an estimate of the population mean. When a random sample is taken from a population, each sample can have a slightly different mean, but the idea is to get an average value that closely represents the true mean of the entire population. For example, in a study with 60 random items that resulted in a sample mean of 80, this value is a point estimate used to infer information about the overall population's average from which the items were taken.

The accuracy of the sample mean as a representation of the population mean depends on several factors, including sample size and variability within the data. As such, statisticians often use the sample mean as the center of a confidence interval to express the reliability of the estimate.

Population Standard Deviation

The population standard deviation, denoted as \( \sigma \), measures the spread of the data points in a population around the mean. It tells us how much, on average, each data point differs from the population mean. A smaller standard deviation indicates that the values tend to be close to the mean, whereas a larger standard deviation suggests a wider spread of values.

In the given example, the population standard deviation is known to be 15. This value is crucial when calculating confidence intervals because it directly influences the margin of error. A larger population standard deviation results in a wider confidence interval for the same level of confidence and sample size, reflecting greater uncertainty about the population mean estimate.

Z-Score

The Z-score, also known as the standard 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 from the mean. In the context of confidence intervals, the Z-score relates to the confidence level. It corresponds to how many standard deviations an element is away from the mean if the data follows a normal distribution.

For example, to calculate a 95% confidence interval, one would look at the Z-score that encompasses 95% of the data in a standard normal distribution. This Z-score is typically referred to as the critical value, and for our case, we use 1.96, which relates to the 95% confidence level. This critical value is then used to multiply by the standard error to calculate the margin of error for a confidence interval.

Sample Size Effect

The effect of sample size on statistical estimates is significant. A larger sample size generally results in a smaller standard error, which means that the sample mean is likely to be a more accurate estimate of the population mean.

This is because the larger the sample, the more likely it is to reflect the true characteristics of the population, reducing the sampling error. In the given exercise, we can see this effect when comparing the margin of error for different sample sizes; for a sample size of 60, the margin of error was larger compared to a sample of 120. The reduction in the margin of error with an increased sample size results in a narrower, and hence, more precise confidence interval.

Margin of Error

The margin of error represents half the width of a confidence interval and reflects the extent of uncertainty or variability around the sample mean estimate of a population mean. It is determined by the critical value (Z-score), the population standard deviation, and the sample size.

The margin of error is calculated using the formula: \( E = Z \times \frac{\sigma}{\sqrt{n}} \), where \( Z \) is the Z-score corresponding to the desired confidence level, \( \sigma \) is the population standard deviation, and \( n \) is the sample size. For example, with a confidence level of 95%, a Z-score of 1.96, a standard deviation of 15, and a sample size of 60, the margin of error is approximately 3.82. This value indicates that the sample mean is likely to differ from the actual population mean by this amount or less.