1. A random sample of n = 16 scores is selected from a normal distribution with a mean of µ = 50 and a standard deviation of á = 10.
What is the probability that the sample mean will have a value between 45 and 55?
What is the probability that the sample mean will have a value between 48 and 52?
What range of values has a 95% probability of containing the sample mean?
Z = (score-mean)/SEm (Standard Error of the mean)
SEm = SD/√(n-1) (You can use n)
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the various Z scores.
95% = mean ± 1.96 SEm
To find the probability that the sample mean will have a value between two given numbers, we can use the Central Limit Theorem. According to the Central Limit Theorem, the distribution of sample means will be approximately normally distributed, regardless of the shape of the original population distribution, as long as the sample size is sufficiently large.
In this case, the sample size is 16, which is considered sufficiently large to apply the Central Limit Theorem.
1. To find the probability that the sample mean will be between 45 and 55, we can calculate the z-scores for these values and then use the z-table to find the corresponding probabilities.
The z-score formula is: z = (x - µ) / (σ / sqrt(n))
Where:
x = given value
µ = population mean
σ = population standard deviation
n = sample size
So for 45:
z1 = (45 - 50) / (10 / sqrt(16)) = -2
And for 55:
z2 = (55 - 50) / (10 / sqrt(16)) = 2
Now we can use the z-table or a statistical calculator to find the area under the standard normal curve between -2 and 2. This gives us the probability that the sample mean will fall between 45 and 55.
2. Similarly, to find the probability that the sample mean will be between 48 and 52, we calculate the z-scores for these values:
For 48:
z1 = (48 - 50) / (10 / sqrt(16)) = -0.8
And for 52:
z2 = (52 - 50) / (10 / sqrt(16)) = 0.8
Using the z-table or a statistical calculator, we find the area under the standard normal curve between -0.8 and 0.8 to get the probability.
3. To find the range of values that has a 95% probability of containing the sample mean, we need to find the z-scores that enclose 95% of the area under the standard normal curve. This is typically done by finding the z-scores for the upper and lower tails that leave 2.5% of the area in each tail.
Using the z-table or a statistical calculator, we can find the z-scores that correspond to the upper and lower tails of 2.5% each. Let's call these z1 and z2.
Then, we can use the z-score formula backwards to find the corresponding values for the sample mean:
For the lower value:
z1 = -1.96 (from the z-table)
x1 = µ + (z1 * (σ / sqrt(n)))
For the upper value:
z2 = 1.96 (from the z-table)
x2 = µ + (z2 * (σ / sqrt(n)))
Therefore, the range of values that has a 95% probability of containing the sample mean is between x1 and x2.