for a normal population with m= 100 and s= 20, For a sample of n= 25 scores, what is the probability that the sample mean will be between 105 & 110?
To find the probability that the sample mean falls between 105 and 110, we first need to calculate the standard error of the sample mean.
The standard error of the sample mean is given by the formula: σ / sqrt(n), where σ is the population standard deviation and n is the sample size.
In this case, σ = 20 (as given) and n = 25 (as given), so the standard error of the sample mean is: 20 / sqrt(25) = 20 / 5 = 4.
Next, we need to standardize the sample mean using the z-score formula: z = (x - μ) / SE, where x is the given value (105 or 110), μ is the population mean (100), and SE is the standard error of the sample mean (4).
For 105: z = (105 - 100) / 4 = 5 / 4 = 1.25
For 110: z = (110 - 100) / 4 = 10 / 4 = 2.5
Now, we need to find the cumulative probability for these z-scores using a standard normal distribution table or calculator.
P(105 < x < 110) = P(1.25 < z < 2.5)
Using a standard normal distribution table, we can find the cumulative probability corresponding to the z-scores:
P(1.25 < z < 2.5) = P(z < 2.5) - P(z < 1.25)
Consulting the standard normal distribution table, we can find that P(z < 2.5) = 0.9938 and P(z < 1.25) = 0.8944.
P(1.25 < z < 2.5) = 0.9938 - 0.8944 = 0.0994
Therefore, the probability that the sample mean will be between 105 and 110 is approximately 0.0994, or 9.94%.