Question

Corn plants from a test plot are measured, and the distribution of heights at \(10-\mathrm{cm}\) intervals is recorded in the following table: $$\begin{array}{cc}\text { Height }(\mathrm{cm}) & \text { Plants (no.) } \\\100 & 20 \\\110 & 60 \\\120 & 90 \\\130 & 130 \\\140 & 180 \\\150 & 120 \\\160 & 70 \\\170 & 50 \\\180 & 40\end{array}$$ 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?

Short answer

Short Answer: Plant heights in the given population have an approximate mean of 135.6 cm and a standard deviation of 23.47 cm. The standard error of the mean is 0.80 cm, suggesting an accurate estimation of the mean height for the entire population. By sketching a plot of plant height against frequency, we can observe if the distribution is approximately normal. Based on the calculated statistics and the shape of the distribution, we can conclude that there is a moderate amount of variation in height within this corn plant population.

Step-by-step solution

Calculate the Mean Height

To find the mean height, we need to find the sum of the product of each height and the number of plants at that height, then divide by the total number of plants. Using the given table, we can calculate the mean height \((\bar{x})\) using the formula: \(\bar{x} = \cfrac{\sum{(x \cdot f)}}{\sum{(f)}}\) $$ \bar{x} = \cfrac{(100\cdot20)+(110\cdot60)+(120\cdot90)+(130\cdot130)+(140\cdot180)+(150\cdot120)+(160\cdot70)+(170\cdot50)+(180\cdot40)}{20+60+90+130+180+120+70+50+40} $$ $$ \bar{x} = \cfrac{2000+6600+10800+16900+25200+18000+11200+8500+7200}{870} $$ $$ \bar{x} \approx 135.6 \ \text{cm} $$

Calculate the Variance

The variance \((s^2)\) can be calculated using the formula: \(s^2 = \cfrac{\sum{(x^2 \cdot f)}}{\sum{(f)}} - (\cfrac{\sum{(x \cdot f)}}{\sum{(f)}})^2\) $$ s^2 = \cfrac{(100^2 \cdot 20)+(110^2 \cdot 60)+(120^2 \cdot 90)+(130^2 \cdot 130)+(140^2 \cdot 180)+(150^2 \cdot 120)+(160^2 \cdot 70)+(170^2 \cdot 50)+(180^2 \cdot 40)}{870} - (135.6)^2 $$ $$ s^2 = \cfrac{200000+726000+1296000+2197000+3528000+2700000+1792000+1445000+1166400}{870} - 18362.56 $$ $$ s^2 \approx 550.9 $$

Calculate the Standard Deviation

The standard deviation \((s)\) is the square root of the variance: $$ s = \sqrt{s^2} = \sqrt{550.9} \approx 23.47 \ \text{cm} $$

Calculate the Standard Error of the Mean

Standard error of the mean (SEM) can be calculated using the formula: \(\text{SEM} = \cfrac{s}{\sqrt{N}}\) For our data, \(N = \sum{(f)} = 870\), so: $$ \text{SEM} = \cfrac{23.47}{\sqrt{870}} \approx 0.80 \ \text{cm} $$

Sketch the Plot of Plant Height Against Frequency

To create a rough sketch of the data, we can plot the given heights (x-axis) against the number of plants at each height (y-axis). The shape of this plot will help us determine if the data follows a normal distribution.

Determine Normality of the Distribution

By inspecting the plotted graph, we can observe if the data follows an approximate bell shape symmetric about the mean value. If the graph shows these characteristics, we can infer that the data distribution is approximately normal.

Assess the Variation Within the Population

In order to assess the variation within the population, we can examine the calculated statistics. The standard deviation (about \(23.47 \ \text{cm}\)) and the standard error of the mean (about \(0.80 \ \text{cm}\)) can provide insight on how much deviation there is from the mean and the accuracy of the mean estimation for the entire population. The standard deviation indicates a moderate amount of variability in plant heights; meanwhile, the standard error implies that the mean height is an accurate measure of the whole population. By carefully considering these statistics and the distribution's shape, an assessment of the variation within this corn plant population can be concluded.

Key concepts

Variance Calculation

Variance is a measure used to describe the spread of data points in a dataset. It helps us understand how much individual data points differ from the mean. To calculate variance, we first need to determine the squared deviations of each height from the mean, then average these over the number of data points.

The formula for variance (\(s^2\)) in this context is:\[s^2 = \frac{\sum{(x^2 \cdot f)}}{\sum{(f)}} - \left(\frac{\sum{(x \cdot f)}}{\sum{(f)}}\right)^2\]Where:

• \( x \) is the height in centimeters.
• \( f \) is the frequency or the number of plants at each height.

The first term calculates the average of the squares of the heights, while the second term adjusts for the squared mean, effectively capturing the dispersion or spread around the mean height. In our exercise, the variance was calculated to be approximately \(550.9\) cm², indicating the variability in the plants' heights.

Standard Deviation Calculation

Standard deviation is a key concept that quantifies the amount of variation or dispersion in a set of data values. It's the square root of the variance and provides a measure that is in the same units as the data itself.

In this exercise, the standard deviation (\(s\)) can be calculated by taking the square root of the already computed variance:\[s = \sqrt{s^2} = \sqrt{550.9} \approx 23.47 \text{ cm}\]This value tells us that the typical deviation from the mean height (which was computed earlier as 135.6 cm) is around 23.47 cm. Thus, most plants fall within this range from the mean, illustrating how tightly or loosely data points are clustered around the mean.
The smaller the standard deviation, the closer the data points are to the mean.

Normal Distribution Assessment

Assessing whether a dataset follows a normal distribution involves checking if the data forms a bell-shaped curve when plotted. This type of distribution is symmetric about the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean.

In the given exercise, we investigate normality by plotting the frequency of each height. If the plot resembles a bell curve, it suggests a normal distribution. Key indicators include:

• Symmetry around the central mean value (135.6 cm).
• Most data points fall within one standard deviation of the mean.
• Follow the empirical rule: approximately 68% of data within one standard deviation, 95% within two, and 99.7% within three standard deviations from the mean.

If these conditions are met, we can reasonably conclude the data is normally distributed, allowing us to apply further statistical analyses effectively.