The numbers below represents heights (in feet) of 3-year old elm trees, where leaves represent fractional parts of each value.
Stems Leaves
5 1,8
6 1,2,4,7,8,9
7 0,2,3,3,4,5
8 1,1,2,3,5,6,6,7,7,9,9
9 0,1,3,4
a) Find the five number summary for the given set of data.
b) Produce a box and whisker plot for the data.
c) Assuming that the heights of 3-year old elm trees are normally distributed find a 95% confidence interval for the mean height of 3-year old elm trees.
a) To find the five number summary for the given set of data (heights of 3-year old elm trees), we need to find the minimum value, maximum value, median (Q2), and the lower quartile (Q1) and upper quartile (Q3).
1) Minimum value: In this case, the minimum value is 5 (the smallest value among all the given heights).
2) Maximum value: The maximum value is 9 (the largest value among all the given heights).
3) Median (Q2): The median (Q2) is the middle value when the data is arranged in ascending order. To find the median, we need to first arrange all the heights in ascending order:
5, 6, 6, 7, 8, 9
Since there is an even number of values, we take an average of the two middle values to find the median:
Median = (6 + 7)/2 = 6.5
4) Lower Quartile (Q1): The lower quartile (Q1) is the median of the first half of the data. In this case, the first half of the data consists of the values:
5, 6, 6
To find the median of this set, we repeat the process:
Q1 = (5 + 6)/2 = 5.5
5) Upper Quartile (Q3): The upper quartile (Q3) is the median of the second half of the data. In this case, the second half of the data consists of the values:
7, 8, 9
Again, we find the median of this set:
Q3 = (7 + 8)/2 = 7.5
So, the five number summary for the given set of data is: Minimum = 5, Q1 = 5.5, Median = 6.5, Q3 = 7.5, Maximum = 9.
b) To produce a box and whisker plot for the data:
- Draw a number line with the minimum value (5) on the left and the maximum value (9) on the right.
- Mark the positions of Q1, Median, and Q3 on the number line.
- Draw a box from Q1 to Q3, enclosing the median.
- Draw a line (whisker) from the left of the box to the minimum value and another line from the right of the box to the maximum value.
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|---------|----------
5 6.5 7.5 9
c) Assuming that the heights of 3-year old elm trees are normally distributed, to find a 95% confidence interval for the mean height of 3-year old elm trees, we need to use the following formula:
Confidence Interval = Mean +/- (Critical Value * Standard Deviation / √(Sample Size))
Since we don't have the population data, we cannot calculate the standard deviation. However, if we assume that the given data is a representative sample, we can make an estimate of the standard deviation.
Using the given data to estimate the standard deviation, we can calculate the sample mean (x̄) by finding the average of all the heights:
x̄ = (5 + 6 + 6 + 7 + 8 + 9) / 6 = 41 / 6 ≈ 6.83
Next, we calculate the sample standard deviation (s) using the sample mean:
s = √((Σ(xi - x̄)²) / (n - 1))
Let's calculate the sample standard deviation step by step:
(xi - x̄) = (5 - 6.83), (6 - 6.83), (6 - 6.83), (7 - 6.83), (8 - 6.83), (9 - 6.83)
(xi - x̄)² = 2.76, 0.7236, 0.7236, 0.0264, 1.8724, 6.7524
Σ(xi - x̄)² = 12.8656
Now, calculate the sample standard deviation:
s = √(12.8656 / (6 - 1)) = √(12.8656 / 5) ≈ √2.57312 ≈ 1.61
Now, to calculate the 95% confidence interval for the mean height of 3-year old elm trees, we need to find the critical value associated with a 95% confidence level. The critical value can be found from the t-distribution table or using statistical software. For simplicity, let's assume the critical value to be 2.
Confidence Interval = 6.83 +/- (2 * 1.61 / √6)
Confidence Interval = 6.83 +/- (2 * 1.61 / √6)
Confidence Interval ≈ 6.83 +/- (3.22 / 2.4495)
Confidence Interval ≈ 6.83 +/- 1.315
So, the 95% confidence interval for the mean height of 3-year old elm trees is approximately (5.515, 8.145) feet.