The heights of the buildings are normally distributed. The mean of the heights is 275 feet and the standard deviation is 30 feet.
A. What is the probability that the buildings height is over 280 feet?
B. What is the probability that a sample of 80 buildings is gathered and the mean height of the buildings was over 280 feet?
a. http://davidmlane.com/hyperstat/z_table.html
b. sample standard deviation= sqrt(30^2/80)= 3.35 recompute, given mean 275, sigma 3.35
To solve these problems, we will use the properties of the normal distribution.
A. What is the probability that the building's height is over 280 feet?
To find this probability, we will use the cumulative distribution function (CDF) of the normal distribution. We want to find the probability that the height is greater than 280 feet.
To calculate this, we need to standardize the value using the formula:
Z = (X - μ) / σ
Where:
- X is the value we want to standardize (280 feet in this case)
- μ is the mean (275 feet)
- σ is the standard deviation (30 feet)
Plugging in the values, we get:
Z = (280 - 275) / 30 = 0.1667
Next, we need to find the area under the standard normal curve to the right of this standardized value. We can do this by looking up the Z-score in the z-table or using a calculator.
Using a z-table, we find that the probability to the right of Z = 0.1667 is approximately 0.4332.
Therefore, the probability that the building's height is over 280 feet is approximately 0.4332.
B. What is the probability that a sample of 80 buildings is gathered, and the mean height of the buildings was over 280 feet?
In this case, we need to consider the sampling distribution of the mean. Since the sample size is large (n = 80), we can assume that the sampling distribution of the mean will be approximately normal.
The mean of the sampling distribution of the mean is equal to the population mean, which is 275 feet. The standard deviation of the sampling distribution of the mean is equal to the population standard deviation divided by the square root of the sample size, i.e., σ / √n, where σ is the population standard deviation (30 feet), and n is the sample size (80).
To find the probability that the mean height of the buildings sampled is over 280 feet, we will use the same method as in part A but with the mean and standard deviation of the sampling distribution of the mean.
The standardized value (Z) can be calculated as follows:
Z = (X - μ) / (σ / √n)
= (280 - 275) / (30 / √80)
= 0.3333
Again, we'll use a z-table or calculator to find the probability to the right of Z = 0.3333.
Using a z-table, we find that the probability to the right of Z = 0.3333 is approximately 0.3707.
Therefore, the probability that a sample of 80 buildings is gathered, and the mean height of the buildings is over 280 feet is approximately 0.3707.