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The standard error is an important concept when dealing with statistics, especially when calculating confidence intervals. It measures the precision of the sample mean as an estimate of the population mean. The formula for the standard error (SE) is the standard deviation of your sample divided by the square root of the sample size, represented as

• \( SE = \frac{s}{\sqrt{n}} \)

Here, \(s\) is the sample standard deviation, and \(n\) is the sample size.
The smaller the standard error, the more precise your sample mean is as an estimate of the population mean. In simpler terms, a small standard error indicates that the sample mean is close to the actual population mean. To calculate it with our numbers, use \(s = 11.9\) and \(n = 53\). This gives:

• \( SE = \frac{11.9}{\sqrt{53}} \approx 1.634 \)

Thus, our confidence interval's precision relies heavily on the standard error.

The Z-score is crucial when you're constructing confidence intervals. It's a statistical measurement describing a value's relationship to the mean of a group of values in terms of standard deviations. To find a Z-score corresponding to the desired confidence level, like a 90% confidence interval, you can look it up in the standard normal distribution table. For a 90% confidence interval, the Z-score is usually around 1.645.
This means that the range spans 1.645 standard deviations away from the mean in both directions. It's this value that scales the standard error in the confidence interval formula. Just remember:

• The higher the confidence level, the larger the Z-score.
• A Z-score helps determine how extreme a value is.

By using a Z-score of 1.645, you can be 90% confident that the true population mean lies within the calculated interval.

The sample mean \(\bar{x}\) is the arithmetic average of a set of sample values. It provides a point estimate of the population mean \(\mu\). The formula to calculate the sample mean is straightforward:

• \( \bar{x} = \frac{\sum x_i}{n} \)

where \(x_i\) are the sample observations and \(n\) is the sample size.
In our exercise, the sample mean is given as \( \bar{x} = 87.2 \). This value acts as the midpoint for our confidence interval, where the interval extends above and below this mean. The sample mean is central because it's the best single estimate we have of the population mean from finite sample data.
Remember:

• The sample mean is sensitive to extreme values or outliers.
• Ensuring a representative sample bolsters the validity of your sample mean.

Understanding the sample mean is vital, as it heavily influences the final confidence interval estimation.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set.

• Conversely, a high standard deviation indicates a wider spread of data.

Standard deviation (\(s\)) is calculated as follows:

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

Here, \(x_i\) are the individual data points, \(\bar{x}\) is the sample mean, and \(n\) is the number of observations.
In the exercise, the standard deviation is \(s = 11.9\), which reflects the variability of the data points from the mean. It's a crucial component when calculating the standard error and hence affects the width of the confidence interval.
Key points:

• Smaller standard deviations lead to narrower confidence intervals.
• Higher standard deviations result in broader intervals.

Standard deviation is key to understanding the reliability and precision of the calculated confidence interval.