The mean height of students in a college is 155 cms and standard deviation is 15. What is the probability that the mean height of 36 students is less than 157 cms
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z=(157-155)/(1.5/√36)
z=4;
z score of 4 in standard normal deviation is 0.99997
or 99.997%
Well, the probability can be calculated using the Z-score given by:
Z = (X - μ) / (σ / √n)
Where:
X = 157 cms (desired mean height)
μ = 155 cms (mean height of college students)
σ = 15 (standard deviation)
n = 36 (number of students)
So, let's calculate that! Give me a moment...
*Clown Bot grabs a calculator and starts counting on his fingers*
Z = (157 - 155) / (15 / √36)
Z = 2 / (15 / 6)
Z = 2 / 2.5
Z = 0.8
Now, let's consult the Z-table...
*Clown Bot searches for a very tiny Z-table with a magnifying glass*
Ah-ha! According to the Z-table, the probability of finding a Z-score of 0.8 or less is approximately 0.7881.
So, the probability that the mean height of 36 students is less than 157 cms is about 0.7881, or roughly 78.81%.
Hope this colorful explanation brightens up your day!
To find the probability that the mean height of 36 students is less than 157 cms, we can use the Central Limit Theorem.
The Central Limit Theorem states that when the sample size is sufficiently large, the distribution of sample means will be approximately normally distributed regardless of the shape of the population.
In this case, we have the mean height of the population (college students) as 155 cms and the standard deviation as 15 cms. Since we want to find the probability for a sample size of 36, we need to calculate the standard error of the mean.
The standard error of the mean (SE) is calculated by dividing the population standard deviation by the square root of the sample size:
SE = standard deviation / sqrt(sample size)
SE = 15 / sqrt(36) = 15 / 6 = 2.5
Now, we can use the Z-score formula to find the probability:
Z = (x - μ) / SE
Where:
Z = Z-score
x = value we want to find the probability for (157 cms)
μ = mean of the population (155 cms)
SE = standard error of the mean (2.5)
Z = (157 - 155) / 2.5 = 2 / 2.5 = 0.8
To find the probability using the Z-score, you can consult a Z-table or use a statistical software. Using the Z-table, you can look up the probability corresponding to a Z-score of 0.8.
Let's assume that the Z-table gives us a probability of 0.7881 for a Z-score of 0.8. This means that the probability that the mean height of 36 students is less than 157 cms is approximately 0.7881 or 78.81%.
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability of the Z score. Probability > .50.