(a) To calculate the mean TSH level, we first need to find the midpoint of each interval and multiply it by the number of people in that interval. Then, we sum up these products and divide by the total number of people.

Midpoints:
2.5 (for 2.0-3.0), 3.5 (for 3.0-4.0), 4.5 (for 4.0-5.0), 5.5 (for 5.0-6.0), 6.5 (for 6.0-7.0), 7.5 (for 7.0-8.0)

Sum of products:
2.5*19 + 3.5*23 + 4.5*31 + 5.5*42 + 6.5*27 + 7.5*18

Divide this sum by the total number of people (160) to get the mean.

To calculate the standard deviation, subtract the mean from each midpoint, square the result, multiply by the number of people in that interval, and sum up these products. Divide this sum by the total number of people minus 1, and then take the square root.

(b) To derive the cumulative frequencies, add up the number of people as you move up the intervals. For the Ogive, plot the upper limit of each interval on the x-axis and the corresponding cumulative frequency on the y-axis, and connect the points.

(c) The median is the value corresponding to the 50th percentile, or the halfway point in the cumulative frequency. The interquartile range is the value at the 75th percentile minus the value at the 25th percentile.

(d) For the box plot, the lower whisker is at the minimum value (2.0), the lower box edge is at the 25th percentile, the line inside the box is at the median, the upper box edge is at the 75th percentile, and the upper whisker is at the maximum value (8.0).

Question

(a) To calculate the mean TSH level, we first need to find the midpoint of each interval and multiply it by the number of people in that interval. Then, we sum up these products and divide by the total number of people.

Midpoints: 
2.5 (for 2.0-3.0), 3.5 (for 3.0-4.0), 4.5 (for 4.0-5.0), 5.5 (for 5.0-6.0), 6.5 (for 6.0-7.0), 7.5 (for 7.0-8.0)

Sum of products: 
2.5*19 + 3.5*23 + 4.5*31 + 5.5*42 + 6.5*27 + 7.5*18

Divide this sum by the total number of people (160) to get the mean.

To calculate the standard deviation, subtract the mean from each midpoint, square the result, multiply by the number of people in that interval, and sum up these products. Divide this sum by the total number of people minus 1, and then take the square root.

(b) To derive the cumulative frequencies, add up the number of people as you move up the intervals. For the Ogive, plot the upper limit of each interval on the x-axis and the corresponding cumulative frequency on the y-axis, and connect the points.

(c) The median is the value corresponding to the 50th percentile, or the halfway point in the cumulative frequency. The interquartile range is the value at the 75th percentile minus the value at the 25th percentile.

(d) For the box plot, the lower whisker is at the minimum value (2.0), the lower box edge is at the 25th percentile, the line inside the box is at the median, the upper box edge is at the 75th percentile, and the upper whisker is at the maximum value (8.0).

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