Question

The amount of growth, in a 15-day period, for a population of sunflower plants was found to follow a normal distribution with mean \(3.18 \mathrm{~cm}\) and standard deviation \(0.53 \mathrm{~cm} .^{9}\) What percentage of plants grow (a) \(4 \mathrm{~cm}\) or more? (b) \(3 \mathrm{~cm}\) or less? (c) between 2.5 and \(3.5 \mathrm{~cm} ?\)

Short answer

The percentage of plants grow (a) 4 cm or more is 100% minus the CDF for the Z-score of 4 cm, (b) 3 cm or less is the CDF for the Z-score of 3 cm, and (c) between 2.5 and 3.5 cm is the CDF for the Z-score of 3.5 cm minus the CDF for the Z-score of 2.5 cm.

Step-by-step solution

Understanding the Normal Distribution

A normal distribution is a bell-shaped curve defined by the mean (average) and standard deviation (spread). The percentage of data within a certain range can be found using the cumulative distribution function (CDF) for normal distribution, or commonly referred to as Z-scores. A Z-score represents the number of standard deviations away from the mean a certain measurement is.

Converting to Z-scores

We convert our values of interest into Z-scores using the formula Z = (X - mean) / standard deviation. Calculate the Z-score for each value (4 cm, 3 cm, 2.5 cm, and 3.5 cm) using the provided mean of 3.18 cm and standard deviation of 0.53 cm.

Calculating Z-scores for Part (a)

For 4 cm: Z = (4 - 3.18) / 0.53. This calculation yields the Z-score for the value 4 cm.

Finding the Percentage for Part (a)

Using standard normal distribution tables or a calculator, find the probability corresponding to the Z-score from Step 3. To find the percentage of plants that grow 4 cm or more, subtract this probability from 1, since the tables give the probability of being less than the Z-score.

Calculating Z-scores for Part (b)

For 3 cm: Z = (3 - 3.18) / 0.53. This calculation yields the Z-score for the value 3 cm.

Finding the Percentage for Part (b)

Use the Z-score from Step 5 to find the corresponding probability in the standard normal distribution table. This percentage represents the proportion of plants that grow 3 cm or less.

Calculating Z-scores for Part (c)

For 2.5 cm and 3.5 cm: Calculate the Z-scores similarly to Step 2. Z for 2.5 cm = (2.5 - 3.18) / 0.53 and Z for 3.5 cm = (3.5 - 3.18) / 0.53.

Finding the Percentage for Part (c)

Find the probabilities corresponding to the Z-scores of 2.5 cm and 3.5 cm from the normal distribution table. The percentage of plants that grow between these two values is the difference between these two probabilities.

Key concepts

Z-scores

Understanding Z-scores is crucial for working with normal distributions. A Z-score, also known as a standard score, indicates how many standard deviations an element is from the mean. In simpler terms, it's a way of standardizing values to easily compare them within a distribution. To calculate a Z-score, you use the formula \(Z = \frac{X - \text{mean}}{\text{standard deviation}}\), where \(X\) is the value you want to compare.

For example, if you have a sunflower that grew 4 cm in 15 days, and the average growth is 3.18 cm with a standard deviation of 0.53 cm, the Z-score would be \(Z = \frac{4 - 3.18}{0.53}\). This calculation shows how many standard deviations above or below the mean the sunflower's growth is, giving you an insight into how unusual (or normal) its growth is within the population.

Standard Deviation

The standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. In the context of our sunflower example, a standard deviation of 0.53 cm means that most sunflowers' growth measurements fall within 0.53 cm of the average 3.18 cm.

When calculating Z-scores, the standard deviation is used to determine how far away a value is from the mean in terms of this spread. If we consider a growth value that is exactly one standard deviation away from the mean, its Z-score would be 1 or -1, depending on whether it's above or below the mean.

Cumulative Distribution Function

The cumulative distribution function (CDF) for a normal distribution provides the probability that a variable will take a value less than or equal to a particular number. Essentially, it's the area under the curve of the distribution to the left of that value. Using our sunflower plant growth distribution as an example, the CDF allows us to determine what percentage of sunflower plants are expected to grow up to a certain height.

Utilizing the CDF

To find the percentage of plants that grow 4 cm or more, we use the CDF to find the probability of plants growing less than 4 cm and then subtract this from 100%. It's important to note that the total area under the normal distribution curve equals 1, representing 100% probability.

Probability

In statistics, probability measures the likelihood that an event will occur. It ranges from 0 to 1, where 0 means the event will not occur, and 1 means the event is certain to occur. Probabilities can also be expressed as percentages.

For our sunflower population, probability helps us understand questions like 'What is the chance that a randomly chosen sunflower will grow 3 cm or less?'. We use the Z-score and the CDF of the normal distribution to find this probability. The value obtained from the CDF corresponds directly to the probability of a plant growing up to that height.

The concept of probability is at the heart of interpreting data within the realm of normal distributions, enabling us to make informed predictions and conclusions about our data set.