STATISTICS Daily Practice WITH SOLUTIONS

A normal distribution has a mean of = 40 with sd = 8. If one score is randomly selected from this distribution, what is the probability that the score will be less than X = 34?

ANSWER .2266 Step 1: Plot mean 40, sd 8, and score of 34. Step 2: Find z-score: 34-40/8=-.75 Step 3: -.75 (portion of the tail), .2266

What z-score value separates the highest 70% of the scores in a normal distribution from the lowest 30%?

Answer: -0.52 Step 1: "from the lowest 30%" Step 2: Look for 30% on portion in the tail. 0.52 Step 3: Since lowest 30% is below mean it is negative. -0.52

A normal distribution has a mean of m = 40 with s = 10. If a vertical line is drawn through the distribution at X = 55, what proportion of the scores are on the right side of the line?

Answer: .0668 Step 1: z-score = 1.5 Step 2: Table: (portion in the tail) .0668

For a population with m = 40 and sd = 8, what is the z-score corresponding to X = 34?

Answer: .75 Step 1: 34-40/8 = .75 [(x-m)/sd]

What proportion of a normal distribution is located in the tail beyond z = 1.50?

Answer: 0.0668 Step 1: located in the tail beyond z = 1.50 Step 2: Table (portion of the tail) 1.50 = 0.668

A normal distribution has a mean of m = 70 with s = 10. If one score is randomly selected from this distribution, what is the probability that the score will be greater than X = 82?

Answer: 0.1151 Step 1: Find z-score for 82. 82-70/10=1.2 Step 2: Look at question: what is the probability that the score will be GREATER than x=82. So it will be in the portion of the tail on the table. Step 3: Table (portion of the tail). 0.115

A normal distribution has a mean of mean = 70 with sd = 10. If one score is randomly selected from this distribution, what is the probability that the score will be greater than X = 82?

Answer: 0.1151 Step 1: Find z-score for 82. 82-70/10=1.2 Step 2: Table (portion of the tail), .1151

A normal distribution has a mean of m = 40 with s = 8. If one score is randomly selected from this distribution, what is the probability that the score will be less than X = 34?

Answer: 0.226 Step 1: Find z-score for 34. 34-40/8=-0.75 Step 2: Table (portion of the tail) .2266

A vertical line drawn through a normal distribution at z = 0.50 separates the distribution into two sections, the body and the tail. What proportion of the distribution is in the tail?

Answer: 0.3085 Step 1: Draw curve and plot .5 with distribution in tail Step 2: Locate .5 on the table (portion of the tail), .3085

John drives to work each morning, and the trip takes an average of 38 minutes. The distribution of driving times is approximately normal with a standard deviation of sd = 5 minutes. For a randomly selected morning, what is the probability that John's drive to work will take between 36 and 40 minutes?

Answer: 0.3108 Step 1: Look at question: Between 36 and 40 minutes Step 2: Plot mean, sd, and 36 & 40 on curve Step 3: Find z-scores for 36 and 40: 36-38/5=-.4; 40-38/5=.4. Step 4: 36 is negative .4, so look at portion of the tail for .4, which is .3446; 40 is above the mean so look to portion in body, which is .6554. Step 5: Subtract: .6554-.3446 = .3108

A vertical line drawn through a normal distribution at z = 0.50 separates the distribution into two sections, the body and the tail. What proportion of the distribution is in the tail?

Answer: 0.385 Step 1: The z-score out be .5 Step 2: On the table for the portion of the tail, answer is .3085

A normal distribution has a mean of m = 70 with s = 12. If one score is randomly selected from this distribution, what is the probability that the score will be less than X = 76?

Answer: 0.6915 Step 1: Find z-score: 76-70/12=.5 Step 2: Table (portion in the body) .6915

What is the probability of randomly selecting a z-score greater than z = -0.75 from a normal distribution?

Answer: 0.7734 Step 1: Draw normal curve showing -.75. Look at question...GREATER than -0.75. Step 2: Find z-score on table for the proportion in the body, .7734.

A sample is selected from a population with population mean = 50 and population sd = 12. If the sample mean of M = 56 produces a z-score of z = +1.00, then how many scores are in the sample?

Answer: 4 Step 1:

Samples of size n = 9 are selected from a population with mean = 80 with sd = 18. What is the standard error for the distribution of sample means?

Answer: 6 Standard error for the distribution of a sample mean: sd of population/sqrt of n. Therefore, 18/sqrt of 9 = 18/3 = 6

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

Answer: 85 Step 1: Plot Mean 100, sd 20, x=80 (z=-1). Step 2: Reason that 90 is -.5 below the mean; and 85 would be -.75 below the mean.

What is the relationship between the alpha level, the size of the critical region, and the risk of a Type I error?

Answer: As the alpha level increases, the size of the critical region increases, and the risk of a Type I error increases.

What position in the distribution corresponds to a z-score of z = -1.00?

Answer: Below the mean by a distance equal to 1 standard deviation

Under what circumstances can a very small treatment effect be statistically significant?

Answer: If the sample size is big and the sample variance is small

Last week, Sarah had exams in math and in Spanish. On the math exam, the mean was = 30 with = 5, and Sarah had a score of X = 45. On the Spanish exam, the mean was = 60 with = 8, and Sarah had a score of X = 68. For which class should Sara expect the better grade?

Answer: Math Step 1: Plot mean, sd, and x value for Math and Spanish. Mark the z-scores for each. Step 2: Z-score for Math is 3; z-score for Spanish is 1. Therefore the higher z-score is the answer (Sarah should expect a better grade).

Why are t statistics more variable than z-scores?

Answer: The extra variability is caused by variations in the sample variance.

A sample of n = 25 scores produces a t statistic of t = -2.062. If the researcher is using a two-tailed test, which of the following is the correct statistical decision?

Answer: The researcher must fail to reject the null hypothesis with either alpha = .05 or alpha = .01.

A normal distribution has a mean of m = 80 with s = 20. What score separates the highest 15% of the distribution from the rest of the scores?

Answer: X = 100.8 Step 1: Look at question and consider 85% is the body. Step 2: Find 85% on table (portion of the body) for z-score of 1.04. Step 3: (z-score x sd) + mean = x (1.04*20)+80 = 100.8

A sample has M = 72 and s = 4. In this sample, what is the X value corresponding to z = -2.00?

Answer: X = 64 Step 1: 72-4=68 (z=-1); 68-4=64 (z=-2)

In a population with sd = 8, a score of X = 44 corresponds to a z-score of z = -0.50. What is the population mean?

Answer: m = 48 Step 1: Problem says: 44 corresponds to -.05. Step 2: Plot normal distribution curve with 0 and -.5 and x=44. Then count. Step 3: Count: (-.5)40 | | | 44 (-.5) | | | 48 (0 mean)

A researcher is using a two-tailed hypothesis test with alpha = .05 to evaluate the effect of a treatment. If the boundaries for the critical region are t = ± 2.080, then how many individuals are in the sample?

Answer: n = 22 Step 1:

A sample is selected from a population with population mean = 70, and a treatment is administered to the sample. After treatment, the sample mean is M = 74, and Cohen's d is d = 1.00. What is the value of the sample variance?

Answer: s2 = 16 Step 1:

For a population with a standard deviation of = 10, what is the z-score corresponding to a score that is 5 points below the mean?

Answer: z = -0.50 Look at question: points BELOW the mean, negative

A sample of n = 9 scores is obtained from a population with m = 70 and sd = 18. If the sample mean is M = 76, what is the z-score for the sample mean? Selected Answer: z = 1.00

Answer: z = 1.00 Step 1: Looking for the z-score of the sample mean. Step 2: Find standard error [sd/sqrt(n)] 18/sqrt. 9 OR 18/3 = 6 Step 3: Find z-score of sample mean: [M-pop. mean/standard error] 76-70/6 = 6/6 = 1

What is the formula for the z-score for the sample mean?

Sample mean (M) - pop. mean/standard error of the sample mean [sd/sqrt(n)]

Samples of size n = 9 are selected from a population with mean = 80 with sd = 18. What is the standard error for the distribution of sample means?

Selected Answer: 6 Step 1: Formula for standard error for distribution of sample mean is [sd/sqrt of n]. Step 2: 18/sqrt. 9= 18/3 = 6

A researcher selects a sample from a population with population mean = 30 and uses the sample to evaluate the effect of a treatment. After treatment, the sample has a mean of M = 32 and a variance of s2 = 6. Which of the following would definitely increase the likelihood of rejecting the null hypothesis?

Selected Answer: All of the other options will increase the likelihood of rejecting the null hypothesis.

A sample of n = 16 scores is obtained from a population with mean = 50 and sd = 16. If the sample mean is M = 54, then what is the z-score for the sample mean?

Selected Answer: z = 1.00 Step 1: [sd/sqrt of n] 16/sqrt 16 = 4 Step 2: Sample mean - population mean/[sd/sqrt of n] Step 3: 54-50/4 = 1

For a population with µ = 80 and σ = 20, the distribution of sample means based on n = 16 will have an expected value of ____ and a standard error of ____.

Step 1: Population: µ = 80, σ = 20 Sample: n = 16 Question: sample expected value is ___ with a standard error of ___ Step 2: mean of population and sample are the same. 80 is the expected value. Step 3: σ/sqrt(n) 20/sqrt(16) 20/4 = 5 5 is the standard error

A population has m = 50. What value of σ would make X = 55 an extreme value out in the tail of the distribution?

s = 1 Because in order to be an extreme value, there would have to be a lot of points.

What is the the formula for the standard error for the distribution of sample means?

sd of population/sqrt of n

A car manufacturer produced 5,000 cars for a limited edition model. Dealers sold all of these cars at mean price of $36,000 with a standard deviation of $3,000. Suppose we were to take random samples of 9 of these cars and calculate the sample mean price for each sample. Calculate the mean and standard deviation of the sampling distribution of x-bar

step 1: population mean and sample mean are always equal. step 2: Population: N = 5000, mean = 36000, sd = 3,000 Sample: n=9 Step 3: population sd/sqrt of sample mean = standard deviation of the sampling distribution Step 4: 3000/sqrt of 9 = 1000

A sample with M = 85 and s = 12 is transformed into z-scores. After the transformation, what are the values for the mean and standard deviation for the sample of z-scores?

σ = 1.00 The SD of a set of standardized scores is always 1