Diane Chen, the Human Resources Manager for a private college, is interested in examining the number of hours overtime worked by staff in one week. She took a sample of 6 staff, with the results given below.3, 6, 4.5, 7, 9, 0.5a) Is the number of hours overtime worked by staff in a week a categorical or numerical variable? If the variable is numerical, determine whether the variable is discrete or continuous. In addition, determine the level of measurement (e.g. ratio or interval scale).b) Calculate the sample mean of this sample (correct to 1 decimal place). Include units.c) Calculate the sample standard deviation (correct to 3 decimal places). Include units.d) What must be assumed about the population distribution in order to calculate a confidence interval? Give a reason for your answere) Using the sample mean and sample standard deviation (found in b and c) calculate the 95% confidence interval for the population mean overtime hours. Include a concluding statement.f) If one staff member claimed they worked 10 hours overtime, based on this sample should you take their statement as true? Why/ why not?Show your working by typing your answer in the text box below.

a) The number of hours overtime worked by staff in a week is a numerical variable because it can be measured in numbers. It is a continuous variable because the number of hours can take on any value within a certain range, not just whole numbers. The level of measurement is ratio because there is a true zero point (no hours worked).

b) To calculate the sample mean, you add up all the values and divide by the number of values. So, (3 + 6 + 4.5 + 7 + 9 + 0.5) / 6 = 30 / 6 = 5.0 hours.

c) To calculate the sample standard deviation, you first calculate the variance, which is the average of the squared differences from the mean. The differences from the mean are (-2, 1, -0.5, 2, 4, -4.5). Squaring these gives (4, 1, 0.25, 4, 16, 20.25). The average of these is 7.583. The square root of this is the standard deviation, which is 2.753 hours.

d) In order to calculate a confidence interval, we must assume that the population distribution is normal or approximately normal. This is because the formula for a confidence interval is based on the properties of the normal distribution.

e) The 95% confidence interval for the population mean is calculated as the sample mean plus or minus the margin of error. The margin of error is the standard deviation divided by the square root of the sample size, multiplied by the z-score for the desired confidence level (1.96 for 95%). So, the margin of error is 2.753 / sqrt(6) * 1.96 = 2.2. Therefore, the confidence interval is 5.0 ± 2.2 hours. This means we are 95% confident that the true population mean is between 2.8 and 7.2 hours.

f) If a staff member claimed they worked 10 hours overtime, this would be outside the range of the 95% confidence interval. Therefore, based on this sample, we would not take their statement as true. However, this does not definitively prove that they did not work 10 hours overtime; it is possible that they are an outlier or that the sample is not representative of the population.