# Probability Standard Error & Expected Value – Statistics

### Probability, Standard Error, Expected Value – Statistics

The key terms used in these Statistics chapters include Sample, Probability, Population, Standard Error, Expected Value – Statistics

What happens to the expected value of M as sample size increases?​

Sample size does not affect the expected value of mean

If other factors are held constant, which set of sample characteristics is most likely to reject a null hypothesis stating that M = 80?

M = 90 and small sample variance

What happens to the standard error of M as sample size increases?​

Error decreases as sample size increases

If random samples, each with n = 9 scores, are selected from a normal population with µ = 80 and σ = 36, then what is the expected value of the mean of the distribution of sample means?​

The expected value of the mean always equals the population mean

For a particular population, a sample of n = 9 scores has a standard error of 8. For the same population, a sample of n = 16 scores would have a standard error of ____.​

First, we need to find the standard deviation:
Transform the formula:
σM = σ/√n

σ = σM √n = 8 3 = 24

Now we use that same standard deviation (24 in this case) to find the new standard error:

σM = σ/√n

= 24/4

= 6

When n is small (less than 30), how does the shape of the t distribution compare to the normal distribution?
It is flatter and more spread out than the normal distribution.

A researcher conducts a hypothesis test using a sample from an unknown population. If the t statistic has df = 30, how many individuals were in the sample?

#### n = 31

What happens to the expected value of M as sample size increases?​

Sample size does not affect the expected value of mean

If other factors are held constant, which set of sample characteristics is most likely to reject a null hypothesis stating that M = 80?

M = 90 and small sample variance

What happens to the standard error of M as sample size increases?​

Error decreases as sample size increases

If random samples, each with n = 9 scores, are selected from a normal population with µ = 80 and σ = 36, then what is the expected value of the mean of the distribution of sample means?​

The expected value of the mean always equals the population mean

With α = .01, the two-tailed critical region for a t test using a sample of n= 16 subjects would have boundaries of ______.

t = ±2.947

Probability Standard Error & Expected Value

On average, what value is expected for the t statistic when the null hypothesis is true?

#### 0

A sample obtained from a population with σ = 12 has a standard error of 2 points. How many scores are in the sample?​

Transform the formula:
σM = σ/√n

n = (σ / σM)2

= (12/2)2

= 36

If all the possible random samples with n = 36 scores are selected from a normal population with µ = 80 and σ = 18, and the mean is calculated for each sample, then what is the average of all the sample means? ​

##### The average of all sample means is the same as the expected value of the mean, which always equals the population mean

Which combination of factors will produce the smallest value for the standard error?​

A large sample size and small population standard error will lead to small standard error. Refer to the formula of the standard error to see why this is the case.

A random sample of n = 4 scores is obtained from a population with a mean of µ = 80 and a standard deviation of σ = 10. If the sample mean is M = 90, what is the z-score for the sample mean?​

First, we need to find the standard error:
σM = σ/√n

= 10/√4

= 5

Then we can apply the z-score formula:

Z = (M-µ)/ σM

= (90-80)/5

= 2

For a normal population with a mean of µ = 80 and a standard deviation of σ = 10, what is the probability of obtaining a sample mean greater than M = 75 for a sample of n = 25 scores?​

First, we need to find the standard error:
σM = σ/√n

= 10/√25

= 2
Then we can apply the z-score formula:
Z = (M-µ)/ σM

= (=75-80)/2

= -2.5

We know that the z-score is to the left of the mean because it is negative. We want the probability of a sample mean greater than z=-2.5. Therefore, we look for the 2.5 in the “proportion in the body” column.

P=0.9938

For a population with μ=100 and σ=20, what is the X value corresponding to z=-0.75?

#### 85x = μ+zσ

Under what circumstances is a score that is located 5 points above the mean a central value, relatively close to the mean?

When the population standard deviation is much greater than 5

Samples of size n = 9 are selected from a population with μ = 80 with σ = 18. What is the expected value of M, the mean of the distribution of sample means?​

The expected value of the mean always equals the population mean

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