Assume that blood pressure readings are normally distributed with a mean of 123 and a standard deviation of 9.6. If 144 people are randomly selected, find the probability that their mean blood pressure will be less than 125.
z = (125-123)/(9.6/sqrt(144)
z = 2/(9.6/12)
z = 2.5
z-table
1-.0062 =0.9938
Bardzo przydatny wpis, bede tego uzywał w swojej pracy!
Well, let's see here. The mean blood pressure is 123, so it seems like most people are just a little stressed out. But don't worry, your blood pressure should be fine as long as you stay calm and keep asking questions. Now, to find the probability that the mean blood pressure of 144 randomly selected people is less than 125, we can use the Central Limit Theorem.
According to the Central Limit Theorem, the distribution of sample means will be approximately normally distributed, even if the population distribution is not normal. Isn't that handy? So, let's calculate the standard deviation of the sample mean.
The standard deviation of the sample mean, often called the standard error, is calculated by dividing the population standard deviation by the square root of the sample size. In this case, the population standard deviation is 9.6 and the sample size is 144. So, the standard error is 9.6 divided by the square root of 144, which is 0.8.
Now, we can find the z-score, which is a measure of how many standard errors a particular value is away from the mean. In this case, we want to find the probability that the mean blood pressure is less than 125. So, we subtract the mean blood pressure from 125 and divide by the standard error.
(125 - 123) / 0.8 = 2.5
Now, we can use a standard normal distribution table to find the probability associated with a z-score of 2.5. In this table, we can see that the probability associated with a z-score of 2.5 is approximately 0.9938.
So, the probability that the mean blood pressure of 144 randomly selected people is less than 125 is approximately 0.9938. Just stay calm and take deep breaths, everything will be alright!
To find the probability that the mean blood pressure of 144 randomly selected people will be less than 125, we can use the Central Limit Theorem.
According to the Central Limit Theorem, when a large enough sample size is taken from a population, the sampling distribution of the mean will be approximately normally distributed, regardless of the shape of the population distribution.
The mean (μ) and standard deviation (σ) of the sampling distribution can be calculated using the following formulas:
Mean of the sampling distribution (μm) = Mean of the population (μ) = 123
Standard deviation of the sampling distribution (σm) = Standard deviation of the population (σ) / √(Sample Size) = 9.6 / √(144)
Substituting the values into the formula:
σm = 9.6 / √(144) = 9.6 / 12 = 0.8
Now, we want to find the probability that the mean blood pressure (x̄) is less than 125. Since the mean and standard deviation of the sampling distribution have been calculated, we can standardize the value of 125 using the formula:
Z = (x̄ - μm) / σm
Substituting the values:
Z = (125 - 123) / 0.8 = 2 / 0.8 = 2.5
Now, we can use a standard normal distribution table or a calculator to find the probability corresponding to a Z-score of 2.5.
Using a standard normal distribution table, the probability that Z is less than 2.5 is approximately 0.9938.
Therefore, the probability that the mean blood pressure of 144 randomly selected people will be less than 125 is approximately 0.9938.
To find the probability that the mean blood pressure of 144 randomly selected people will be less than 125, we can use the Central Limit Theorem. According to this theorem, when a large enough sample is taken from a population with any distribution, the distribution of the sample means will be approximately normal, regardless of the shape of the population distribution.
To apply the Central Limit Theorem, we need to adjust the mean and standard deviation of the population distribution based on the sample size. The mean of the sample means will be equal to the population mean, which is 123. The standard deviation of the sample means (also known as the standard error) can be calculated by dividing the population standard deviation by the square root of the sample size.
In this case, the sample size is 144, so we calculate the standard error as follows:
Standard error = population standard deviation / √sample size
= 9.6 / √144
= 9.6 / 12
= 0.8
Now we can use this adjusted distribution to find the probability of the mean blood pressure being less than 125.
To do this, we need to find the Z-score associated with a mean blood pressure of 125 in this adjusted distribution. The Z-score represents the number of standard deviations away from the mean a particular value is. We can calculate the Z-score using the formula:
Z = (X - μ) / σ
Where:
Z is the Z-score,
X is the value we want to find the probability for (125 in this case),
μ is the population mean (123 in this case),
and σ is the standard deviation of the sample means (0.8 in this case).
So, substituting the values into the formula, we get:
Z = (125 - 123) / 0.8
= 2.5
Now that we have the Z-score, we can use a Z-table or a calculator to find the corresponding probability. Looking up the Z-score of 2.5 in a Z-table, we find that the probability is approximately 0.9938.
Therefore, the probability that the mean blood pressure of 144 randomly selected people will be less than 125 is approximately 0.9938.