A population distribution is known to have standard deviation 20.
Determine the p value of a test of the hypothesis that the population
mean is equal to 50, if the average of a sample of 64 observations
is a) 52.5 , b) 55.0 , c) 57.5
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 for each Z score.
To determine the p-value of a hypothesis test, we need to calculate the test statistic and compare it to the appropriate distribution. In this case, since the sample size is large (64 observations), we can use the normal distribution.
The test statistic is calculated using the following formula:
z = (x - μ) / (σ / sqrt(n))
Where:
x is the sample mean,
μ is the population mean we're testing against (50),
σ is the population standard deviation (20), and
n is the sample size (64).
a) For x = 52.5:
z = (52.5 - 50) / (20 / sqrt(64))
z = 2.5 / (20 / 8)
z = 2.5 / 2.5
z = 1
b) For x = 55.0:
z = (55.0 - 50) / (20 / sqrt(64))
z = 5 / (20 / 8)
z = 5 / 2.5
z = 2
c) For x = 57.5:
z = (57.5 - 50) / (20 / sqrt(64))
z = 7.5 / (20 / 8)
z = 7.5 / 2.5
z = 3
Now that we have the z-scores, we can look up the p-values associated with these values in the standard normal distribution table. The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. In this case, since we're performing a two-tailed test, we need to find the probability of observing a z-score as extreme or more extreme than the ones calculated.
Using the standard normal distribution table, we find the following p-values:
a) p ≈ 0.1587 (for z = 1)
b) p ≈ 0.0228 (for z = 2)
c) p ≈ 0.0013 (for z = 3)
Therefore, the p-values for the three different sample means are approximately:
a) p ≈ 0.1587
b) p ≈ 0.0228
c) p ≈ 0.0013
To determine the p-value of a hypothesis test, we can use a z-test. The formula for the test statistic (z-score) is:
z = (x - μ) / (σ / sqrt(n))
Where:
- x is the value of the sample mean
- μ is the hypothesized population mean
- σ is the standard deviation of the population
- n is the sample size
In this case, the population standard deviation is known to be 20, and the hypothesized population mean is 50. The sample size is 64.
a) For x = 52.5:
z = (52.5 - 50) / (20 / sqrt(64))
z = 2.5 / (20 / 8)
z = 2.5 / 2.5
z = 1
b) For x = 55.0:
z = (55.0 - 50) / (20 / sqrt(64))
z = 5 / (20 / 8)
z = 5 / 2.5
z = 2
c) For x = 57.5:
z = (57.5 - 50) / (20 / sqrt(64))
z = 7.5 / (20 / 8)
z = 7.5 / 2.5
z = 3
Now, we need to find the p-value associated with each z-score using a standard normal distribution table or a statistical calculator.
a) For z = 1, the p-value is the probability of obtaining a z-score less than 1 or greater than -1. Using a standard normal distribution table, we find the p-value to be approximately 0.6827.
b) For z = 2, the p-value is the probability of obtaining a z-score less than 2 or greater than -2. Using a standard normal distribution table, we find the p-value to be approximately 0.0228.
c) For z = 3, the p-value is the probability of obtaining a z-score less than 3 or greater than -3. Using a standard normal distribution table, we find the p-value to be approximately 0.0013.
Therefore, the p-values for each case are:
a) p ≈ 0.6827
b) p ≈ 0.0228
c) p ≈ 0.0013