In a certain population the heights of men are normally distributed with mean 168.75 and variance 42.25 those of women are normally distributed with mean 162.50 and variance 36 the probability that the mean height of a random sample of 25 men is less than 170 is
To find the probability that the mean height of a random sample of 25 men is less than 170, we can use the Central Limit Theorem.
The Central Limit Theorem states that for a large enough sample size, the distribution of sample means will be approximately normally distributed, regardless of the shape of the population distribution.
In this case, we have a normal population distribution for men's heights with a mean of 168.75 and a variance of 42.25. Since the variance is given, we can calculate the standard deviation by taking the square root of the variance.
Standard deviation (σ) = √Variance = √42.25 ≈ 6.5
Next, we need to find the standard error of the mean (SE) for the sample. The standard error of the mean is equal to the standard deviation divided by the square root of the sample size.
Standard error of the mean (SE) = σ / √sample size = 6.5 / √25 = 6.5 / 5 = 1.3
Now that we have the standard error of the mean, we can calculate the z-score. The z-score represents the number of standard deviations a given value is from the mean.
z = (x - μ) / SE
Where:
x = 170 (the value we want to find the probability for)
μ = 168.75 (mean of the population)
SE = 1.3 (standard error of the mean)
z = (170 - 168.75) / 1.3 ≈ 0.96
To find the probability, we can look up the corresponding z-score in a standard normal distribution table or use a calculator or software that provides the cumulative distribution function (CDF) for the standard normal distribution.
Using a standard normal distribution table or calculator, we find that the probability corresponding to a z-score of 0.96 is approximately 0.8315.
Therefore, the probability that the mean height of a random sample of 25 men is less than 170 is approximately 0.8315 or 83.15%.
To find the probability that the mean height of a random sample of 25 men is less than 170, we can use the central limit theorem.
According to the central limit theorem, when the sample size is large enough (typically greater than 30) regardless of the population's distribution, the distribution of sample means will be approximately normally distributed.
For the given problem, we are looking for the probability that the mean height of a random sample of 25 men is less than 170.
The mean height of men is given as 168.75, and the variance is given as 42.25. The standard deviation (σ) is the square root of the variance:
σ = √42.25 = 6.5
We need to calculate the standard error (SE) which is the standard deviation of the sample mean. It is calculated by dividing the population standard deviation by the square root of the sample size:
SE = σ / √n
Here, n = 25, so the standard error is:
SE = 6.5 / √25 = 6.5 / 5 = 1.3
Next, we need to standardize the value of 170 using the sample mean and standard error. To calculate the z-score, we use the formula:
z = (x - μ) / SE
Here, x = 170, μ = 168.75 (mean height of men), and SE = 1.3 (standard error):
z = (x - μ) / SE
z = (170 - 168.75) / 1.3
z = 0.96
Finally, we need to find the probability corresponding to this z-score using the normal distribution table or a calculator. The probability can be found by looking up the z-score in the table or using calculator functions, which gives us:
P(Z < 0.96) = 0.8315
Therefore, the probability that the mean height of a random sample of 25 men is less than 170 is approximately 0.8315, or 83.15%.