### 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/√nσ = σM √n = 8 3 = 24Now 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/√nn = (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= 5Then 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= 2Then we can apply the z-score formula:Z = (M-µ)/ σM= (=75-80)/2= -2.5We 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?

**85**

**x = 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**