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The sample mean is a foundational concept in statistics, representing the average of a collection of numbers. In this exercise, the numbers represent the count of emails used by seven women aged 80 or over. To calculate the sample mean, all the values in our dataset are added together and then divided by the total number of observations.

The formula for the sample mean (\( \bar{x} \) ) is:

• \( \bar{x} = \frac{\text{Sum of all observations}}{\text{Total number of observations}} \)

For our dataset, the numbers are 0, 0, 1, 2, 5, 7, and 14. Summing these numbers gives 29, and dividing by the number of observations, 7, results in a sample mean of approximately 4.14.

Understanding the sample mean helps in summarizing the dataset with just one statistic, providing an indication of the central tendency of the data.

Standard deviation is a key statistic that quantifies how much the values in a dataset differ from the mean. It provides insight into the variability or spread of the dataset. To compute the standard deviation, the deviation of each data point from the mean is squared, summed, and then averaged.

The formula for standard deviation (\( s \) ) is:

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

Here, \( x_i \) represents each data point, \( \bar{x} \) is the sample mean, and \( n \) is the number of observations.

For our data, the deviations from the mean (4.14) are calculated and squared, resulting in a sum of squares of 154.86. The standard deviation is then approximately 5.08.

A higher standard deviation indicates greater variability among the data points, offering insights into how spread out the data is around the mean.

A confidence interval gives a range of values, derived from the sample data, that is believed to contain the population mean. It provides a measure of uncertainty around the sample mean. The 90% confidence interval, for example, suggests that if the same population is sampled multiple times, approximately 90% of the calculated intervals would contain the true population mean.

The formula for a confidence interval (CI) is:

• \( CI = \bar{x} \pm z \times SE \)

Where

• \( \bar{x} \) is the sample mean,
• \( z \) is the z-value from the standard normal distribution for the desired confidence level,
• and \( SE \) is the standard error of the mean.

The standard error (SE) is calculated by dividing the standard deviation by the square root of the sample size.

In this instance, using a 90% confidence level with a z-value of 1.645, the confidence interval for the sample mean of approximately 4.14 is (1.99, 6.28). This interval indicates that we can be 90% confident that the true mean is contained within this range.

A skewed distribution occurs when data points are not symmetrically distributed around the mean, leading to a longer tail on one side. In this context, the dataset is skewed to the right because most values are lower, with a few exceptionally high values (e.g., 14 emails) stretching out towards the right.

Right skewness is common in datasets with natural limits at zero but few high outliers, which influence the mean to be higher than the median. This occurs in various real-world scenarios where values are clustered at one end, with occasional much larger values.

A skewed distribution affects the interpretation of the mean and the confidence interval. In highly skewed distributions, the confidence interval may be less precise, as assumptions about normality are violated. However, the interval still serves as a useful estimation tool. Although caution is warranted in interpretation, especially with small sample sizes, the confidence interval can still offer insights into the population mean.