Question

Corn plants from a test plot are measured, and the distribution of heights at 10-cm intervals is recorded in the following table:
Height (cm)   Plants (no.)
100          20
110          60
120          90
130         130
140         180
150         120
160          70
170         50
180         40
Calculate
(a) the mean height,
(b) the variance,
(c) the standard deviation, and
(d) the standard error of the mean.
Plot a rough graph of plant height against frequency. Do the values represent a normal distribution? Based on your calculations, how would you assess the variation within this population?

Accepted Answer

Step 1: Calculate the total number of plants by summing all the frequencies given for each height interval. This total will be used as the denominator for calculating the mean and other statistics. Step 2: Calculate the mean height using the formula for the weighted average: $$	ext{Mean} = \frac{\sum (x_i 	imes f_i)}{\sum f_i}$$, where $$x_i is $$the height value and $$f_i is $$the number of plants at that height. Step 3: Calculate the variance using the formula: $$	ext{Variance} = \frac{\sum f_i (x_i - 	ext{Mean})^2$}{\sum f_i}. $$This involves subtracting the mean from each height, squaring the result, multiplying by the frequency, summing all these values, and dividing by the total number of plants. Step 4: Calculate the standard deviation by taking the square root of the variance: $$	ext{Standard Deviation} = \sqrt{	ext{Variance}}. $$This gives a measure of the spread of the heights around the mean. Step 5: Calculate the standard error of the mean (SEM) using the formula: $$	ext{SEM} = \frac{	ext{Standard Deviation}}{\sqrt{n}}$$, where $$n is $$the total number of plants. This quantifies the precision of the mean estimate. Step 6: To plot the graph, place plant height on the x-axis and frequency (number of plants) on the y-axis. Connect the points to visualize the distribution shape. Assess whether the distribution resembles a normal distribution by checking if it is roughly symmetric and bell-shaped. Step 7: Based on the calculated variance and standard deviation, discuss the degree of variation within the population. A smaller standard deviation indicates less variation, while a larger one indicates more spread in plant heights.

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Concetti chiave

Measures of Central Tendency (Mean)

Variance and Standard Deviation

Standard Error of the Mean and Normal Distribution