Sampling Distribution
Complete solutions and illustrations are required.
3.The mean systolic blood pressure of normal adults is 120 millimeters of mercury (mm Hg) and the standard deviation is 5.6. Assume that the variable is normally distributed.
a.If an individual is selected, find the probability that the individual’s pressure will be between 118 and 122 mm Hg.
b.If a sample of 35 adults is randomly selected, find the probability that the sample mean will be between 119 and 121 mm Hg.
a. Z = (score-mean)/SD
b. Z = (score-mean)/SEm
SEm = SD/√n
With both problems, find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores.
To calculate the probabilities in these scenarios, we need to use the concept of the sampling distribution of means.
The sampling distribution of means refers to the distribution of all possible sample means that can be obtained from a population. With a large enough sample size, the sampling distribution of means will be approximately normally distributed, regardless of the shape of the population distribution.
In this case, we can assume that the individual blood pressure readings are normally distributed, which means that the sample means will also be normally distributed.
To solve the problems, we can use the Z-distribution and standard normal distribution tables or a statistical software, such as Excel or R, to find the probabilities.
a. Probability of an individual's pressure between 118 and 122 mm Hg:
To calculate this probability, we can use the Z-score formula:
Z = (X - μ) / σ
Where:
X = desired blood pressure value
μ = population mean
σ = population standard deviation
In this case:
X1 = 118 mm Hg
X2 = 122 mm Hg
μ = 120 mm Hg
σ = 5.6
We need to find the probability that Z1 < Z < Z2, where Z1 and Z2 are the Z-scores corresponding to X1 and X2, respectively.
To find the Z-scores, we use the formula:
Z = (X - μ) / σ
Z1 = (X1 - μ) / σ
Z1 = (118 - 120) / 5.6
Z2 = (X2 - μ) / σ
Z2 = (122 - 120) / 5.6
Now, we can use either a standard normal distribution table or a statistical software to find the probabilities associated with these Z-scores.
For example, using a standard normal distribution table, we can find the probability associated with Z1 and Z2 and calculate the difference between them:
P(Z1 < Z < Z2) = P(Z2) - P(Z1)
b. Probability of sample mean between 119 and 121 mm Hg:
To find the probability that the sample mean falls between 119 and 121 mm Hg, we need to calculate the Z-scores for the sample mean values, using the same formulas as above.
Now, we need to find the probability that Z1 < Z < Z2, where Z1 and Z2 are the Z-scores corresponding to 119 and 121, respectively.
Again, we can use a standard normal distribution table or a statistical software to find the probabilities associated with these Z-scores.
To summarize, the steps to solve the problems are as follows:
1. Calculate the Z-scores using the formula Z = (X - μ) / σ
2. Look up the probabilities associated with the Z-scores using a standard normal distribution table or a statistical software.
3. Calculate the desired probabilities by subtracting the values obtained in step 2.
I hope this explanation helps you understand the process of solving these problems using the sampling distribution of means concept.