Final answer:
To find the percentage of plants between 135 and 155 cm tall in a normal distribution with mean 145 cm and standard deviation 23 cm, we can calculate the z-scores for both values and use the standard normal distribution table or a calculator to find the area under the curve between those z-scores. The result is approximately 33.2%.
Step-by-step explanation:
To find the percentage of plants between 135 and 155 cm tall, we need to calculate the z-scores for both values using the formula:
z = (x - mean) / standard deviation
Using the given mean of 145 cm and standard deviation of 23 cm:
For 135 cm: z = (135 - 145) / 23 = -0.43
For 155 cm: z = (155 - 145) / 23 = 0.43
Next, we can use the z-scores to find the percentage of plants between these two heights using the standard normal distribution table or a calculator. The area under the curve between -0.43 and 0.43 is approximately 0.332, or 33.2%.
The Complete Question is:
The height of a certain population of corn plants follow a normal distribution with mean 145 cm and standard deviation 23 cm.
(a) what percentage of the plants are between 135 and 155 cm tall?
(b) suppose we were to choose at a random from the population a large number of samples of 16 plants each. In what percentage of the samples would the sample mean height be between 135 and 155 cm.
(c) if y(bar) represents the mean height of a random sample of 16 plants from the population, what is Pr[135<_Y(bar)<_ 155]
(d) If Y(bar) represents the mean height of a random sample of 36 plants from the population, what is Pr[135<_Y(bar)<_ 155]