Final answer:
To find the percentage of plants between specific heights in a given population with a normal distribution, calculate the z-scores for the heights and use a standard normal distribution table or calculator to find the percentage. To find the percentage of samples with a mean height between specific values, use the distribution of the sample mean with the same mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.
Step-by-step explanation:
a. To find the percentage of plants between 135 and 155 cm tall, we need to calculate the z-scores for these heights using the formula: z = (x - mean) / standard deviation. For 135 cm: z = (135 - 145) / 22 = -0.45. For 155 cm: z = (155 - 145) / 22 = 0.45. We can then use a standard normal distribution table or a calculator to find the percentage of plants between these z-scores. The area under the curve between -0.45 and 0.45 is approximately 0.3311 or 33.11%.
b. To find the percentage of samples with a mean height between 135 and 155 cm, we can use the fact that the sample mean follows a normal distribution with the same mean as the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size. In this case, the population standard deviation is 22 cm and the sample size is 16. So, the standard deviation of the sample mean is 22 / sqrt(16) = 5.5 cm. We can then use the same method as part a to calculate the z-scores for the sample mean and find the percentage of samples between these z-scores.