random sample of size 36 is taken from a population with mean 50 and standard deviation 5. Find P( x¯ < 51.5).
To solve this problem, we can use the Central Limit Theorem and convert it into a standard normal distribution.
The Central Limit Theorem states that the distribution of the sample means approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.
First, we need to calculate the standard deviation of the sample mean (also known as the standard error), which is equal to the population standard deviation divided by the square root of the sample size.
Standard deviation of the sample mean = 5 / √36 = 5 / 6 = 0.8333
Now, we can calculate the z-score, which measures the number of standard deviations a value is away from the mean.
z = (51.5 - 50) / 0.8333 = 1.80
Using a standard normal distribution table, we can find the probability P(x < 1.80):
P(x < 1.80) = 0.9641
Therefore, the probability P(x̄ < 51.5) is approximately 0.9641 or 96.41%.
To find the probability, P(x¯ < 51.5), we can use the Central Limit Theorem, which states that the distribution of sample means approaches a normal distribution as the sample size increases.
Step 1: Calculate the standard deviation of the sample mean.
The standard deviation of the sample mean (also known as the standard error) can be calculated using the formula:
Standard Error = standard deviation / √n
where n is the sample size.
In this case, the standard deviation is 5 and the sample size is 36. So, the standard error is:
Standard Error = 5 / √36 = 5 / 6 = 0.8333
Step 2: Convert the given value (51.5) to a z-score.
The z-score measures how many standard deviations an observation or sample mean is from the mean of the population. It can be calculated using the formula:
z = (x - μ) / σ
where x is the value of interest, μ is the population mean, and σ is the standard deviation of the population or standard error of the sample mean.
In this case, the value of interest is 51.5, the population mean is 50, and the standard error is 0.8333. So, the z-score is:
z = (51.5 - 50) / 0.8333 ≈ 1.80
Step 3: Use a standard normal distribution table or calculator to find the probability.
Using a standard normal distribution table or calculator, we can find the area under the curve to the left of the z-score. In this case, we want to find P(x¯ < 51.5), which is equivalent to finding P(z < 1.80).
Referring to the standard normal distribution table, we can find that the area to the left of 1.80 is approximately 0.9641 or 96.41%.
Therefore, P(x¯ < 51.5) ≈ 0.9641 or 96.41%.
Note: The above calculation assumes that the population distribution is approximately normal, or the sample size is large enough to invoke the Central Limit Theorem.
Z = (51.5-50)/5 = .3
Consult your normal distribution table in the back of your statistics text.