Exam 2
Calculate the test statistic for this test.
-2.5
If we set α = 0.05, what can we do to increase power?
Increase sample size.
Refer to the process described in Question 8. On the basis of the following sample means, can we say that the process is "in control"?
No, because there is a run of 9 or more terms
What is the name of the quantity s/√n
standard error of x-bar
Suppose a sample of size 250 was taken instead of size 100. How will the margin of error change?
the margin of error will decrease in size
Suppose we have a very right skewed population distribution where μ = 80 and σ = 20. For random samples of size n = 100, what is the standard deviation of the sampling distribution of x̄?
Less than 20
Two studies were done on the same set of data, where study I was a one-sided test and study II was a two-sided test. The p-value of the test corresponding to study I was found to be 0.030. What is the p-value for study II?
The p-value must be 0.060
True or False: Statistical inference can be defined as making generalizations about the population based on sample data.
True
What is the probability that any random sample of n = 100 results in an x̄ between 77.3 and 81.1?
0.6203
Suppose we take a sample of size n = 10 from this same population. Can we compute the probability that x̄ is greater than 75?
Yes, because the population is normally distributed. Thus, the sampling distribution of x-bar is normally distributed.
Refer to the previous question. In order to construct an x̄-chart for this process, what number should be used as the center line?
15.0
Consider the control chart above. Is the process out of control?
Yes, there is a run of 9 or more consecutive points on the same side of the center line.
If the researcher wanted to have 95% confidence in the results with a margin of error of 5.1, how many students must be sampled? (Assume σ=15)
34
Your company markets a computerized medical diagnostic program used to evaluate thousands of people. The program scans the results of routine medical tests (pulse rate, blood tests, etc.) and refers the case to a doctor if there is evidence of a medical problem. The program makes a decision about each person. The hypotheses are: H0: The patient is healthy. Ha: The patient has a medical problem. Which of the following describes a Type I error in this situation? A.Not referring a patient with a medical problem to the doctor. B.Sending a patient with a medical problem to the doctor. C.Sending a healthy patient to the doctor. D.Not referring a healthy patient to the doctor.
C. Sending a healthy patient to the doctor.
True or False: If there is not enough evidence to support the alternative hypothesis, we can accept the null hypothesis.
False
True or False: Point estimation is a form of statistical inference in which, based on the sample data, we estimate the unknown parameter of interest using a range of plausible values.
False
True or False: The null hypothesis is the claim that the researcher wants to prove.
False
True or False: We can never compute probabilities on x̄ when the population is skewed.
False
The following sample means were collected by taking random samples of size n = 25. These sample means were computed from samples taken from a Normal population with μ = 52.4 and σ = 7.5. On the basis of these sample means, can we conclude that the process is "in control"?
No, because at least one of the means is either below the lower control limit or above the upper control limit.
With a p-value of 0.287, what is the appropriate conclusion to make?
Our data do not provide strong enough evidence for rejecting H0.
Based on sample results, researchers estimate that μ, the mean height of all female students in a university, is 64 inches. What type of inference is being used?
Point Estimation
Suppose that the p-value was 0.0259. What is the appropriate conclusion to make if α = 0.05?
Reject H0. We have sufficient evidence to conclude that the mean concentration is different from 250 ppm.
Refer to the process described in Question 8. On the basis of the following sample means, can we say that the process is "in control"?
Yes, because all of the means vary between the lower and upper control limits.
All students in the US who took the ACT in 2014 had a mean score of μ=21.0. Suppose you randomly select two samples of students from this population, and you calculate the sample mean for each. Sample 1 has a size of n=40, and Sample 2 has a size of n=250. Which sample is more likely to get a sample mean of 18 or less?
Sample 1 is more likely
A process is being monitored using an x̄-chart. Sample means were computed from random samples of size n = 16 from a Normal population with μ = 35.0 and σ = 5.0. On the basis of the following x̄-chart, can we say that the process is in control?
Yes, because all the x̄'s are between the upper and lower control limits and there are only four in a row above the center line.
Which of the following describes statistical power in this situation?
The probability of sending a patient with a medical problem to the doctor
True or False: Suppose we take all possible samples of the same size from a population and for each sample, we compute x̄. The mean of these x̄ values will be exactly equal to the mean of the population (μ) from which the samples were taken.
True
True or False: Suppose we take all possible samples of the same size from a population and for each, we compute x̄. The standard deviation of these x̄ values will be less than or equal to the standard deviation of the population from which the samples were taken.
True
The IQ level of students at a particular university has an unknown mean. A simple random sample of 100 students is found to have a sample mean IQ of x̄=115 and a sample standard deviation of s=15. Calculate a 95% confidence interval for the mean IQ level of all students in this university.
(112, 118) 115+-2(15/10)
Suppose that the test statistic was -3.00. What is the p-value for this test?
0.002 < p-value < 0.005
Suppose we take a sample of size n = 50 from this same population, and we calculate x̄ = $2,800. How many standard deviations,σ/√ⁿ , away from μ is this sample mean?
2.83
According to the Central Limit Theorem, for random samples, what is the approximate shape of the sampling distribution of x̄ when the population distribution is non-Normal?
Approximately Normal if the sample size is large
Suppose we take a sample of size n = 100 from this same population. Can we compute the probability that x̄ is greater than $2,800?
Yes, because the Central Limit Theorem applies. Thus, the sampling distribution of x-bar is normally distributed.
Suppose we have a very right skewed population distribution where μ = 80 and σ = 20. For random samples of size n = 100, what is the shape of the sampling distribution of x̄?
Approximately Normal
Refer to Question 2. What is the lower limit of the control chart for this process?
14.4
Refer to Question 2. What is the upper limit of the control chart for this process?
15.6
Suppose the p-value for this test is 0.0734. What are the appropriate conclusions to make when α = 0.05?
Fail to reject the null hypothesis. There is insufficient evidence to conclude that the mean hemoglobin level of all children in Jordan is less than 12 g/dl.
A study of child preferences for milk chocolate was performed. Based on sample results, researchers were 95% confident that the proportion of children that liked milk chocolate was between 0.75 and 0.93. What type of inference is being used?
Interval Estimation
Fill in the blank: For small random samples from a Normal population distribution, the shape of the sampling distribution of x̄ is __________.
Normal
Suppose we are testing H0: μ = 10 versus Ha: μ < 10. The upper right hand curve represents the sampling distribution of x̄ when H0 is true; the lower left hand curve represents the sampling distribution of x̄ when Ha is true. What is the color of the area that represents α if μ is really 10?
Red (area to the left of the line on the top curve)
Hemoglobin is a protein in the red blood cells that carries oxygen from the lungs to body tissues. People with less than 12 grams of hemoglobin per deciliter of blood (g/dl) are anemic. A public health official in Jordan suspects that the mean hemoglobin level for all children is less than 12 grams. He took a random sample of 50 children and found x̄ = 11.5 g/dl and s = 1.6 g/dl.
The mean hemoglobin level of all children in Jordan
True or False: If our data give results that are extremely unlikely to occur if H0 were true, then we have strong evidence against H0 and can reject it in favor of Ha.
True
True or False: Increasing the confidence level will lead to a wider margin of error.
True
Suppose we take a random sample of size 75. What is the probability that the mean of the 75 awards is less than $2,550?
0.1922 (2500-2600)/(500/8.66) =-8.72 (look up on Z table) =0.8078 1-0.8078=1.922
Suppose the p-value is 0.0367. What is the correct interpretation of this p-value?
Assuming the null hypothesis is true, there is a 0.0367 probability of obtaining a sample statistic as extreme or more extreme than what we calculated.
True or False: Statistically significant results are always of practical importance.
False
For many years, "working full-time" has meant working 40 hours per week. Nowadays it seems that corporate employers expect their employees to work more than this amount. A researcher decides to investigate this hypothesis. The null hypothesis states that the average time full-time corporate employees work per week is 40 hours. The alternative hypothesis states that the average time full-time corporate employees work per week is more than 40 hours. To substantiate his claim, the researcher randomly selected 250 corporate employees and finds that they work an average of 47 hours per week with a standard deviation of 3.2 hours. In order to assess the evidence, what do we need to ask?
How likely it is that, in a sample of 250, we will find that the mean number of hours per week full-time corporate employees work is as high as 47 if the true mean is 40?
A recent study claimed that half of all college students "drink to get drunk" at least once in a while. Believing that the true proportion is much lower, the College Alcohol Study interviews an SRS of 14,941 college students about their drinking habits and finds that 7,352 of them occasionally "drink to get drunk". What type of statistical inference is this?
Hypothesis Testing
A university administrator obtains a sample of the academic records of past and present scholarship athletes at the university. The administrator reports that the mean GPA (grade point average) of a random sample of 40 male scholarship athletes is 3.02 and the mean GPA of a random sample of 36 female scholarship athletes is 3.11. If there is no difference in the mean GPA of male and female athletes, the probability of obtaining this difference (3.11 - 3.02 = 0.09) or more extreme is approximately 0.287. What is the null hypothesis in this example?
Male and female scholarship athletes have the same mean GPA.
the weight of a carton of a dozen eggs produced by a certain breed of hens is supposed to be normally distributed with a mean of 780 grams. A quality manager randomly checks thirty-five cartons of eggs (n = 35) to see whether the mean weight differs from 780 grams. She finds x̄ = 796 grams and computes a 95% confidence interval estimate for the mean weight of all cartons to be (787.82 grams, 804.18 grams). On the basis of this interval, at α = 0.05, what can she conclude about H0: μ = 780 versus Ha: μ ≠ 780?
Reject H0 since 780 is outside the given interval.
suppose the p-value is 0.0367. At α = 0.05, what should the public health official conclude?
Reject the null hypothesis. The true mean hemoglobin level of all children in Jordan is less than 12 g/dl.
Suppose the calculated test statistic is t = -2.16. Assuming all conditions are met, what is the p-value for this left-tailed test?
0.01 < p-value < 0.02
Suppose we have a normal population distribution where μ = 80 and σ = 20. For random samples of size n = 100, what is the probability of getting an x̄ greater than 75?
0.9938 (80-75)/(20/10)= -2.5 (look up on table) =0.0062 1-0.0062= 0.9938
Which of the following describes a Type II error in this situation? A. Not referring a patient with a medical problem to the doctor. B. Sending a patient with a medical problem to the doctor. C. Sending a healthy patient to the doctor. D. Not referring a healthy patient to the doctor.
A. Not referring a patient with a medical problem to the doctor.
Fill in the blank: The sampling distribution of x̅ gives ___________ from all possible samples of the same size from the same population
All x̅ values
Which of the following is a Parameter? A. The proportion of voters in a national sample of 1500 who say they will vote for the incumbent president B. The mean GPA of all incoming freshmen at a state university C. The standard deviation of the weights of a sample of National League Football players D. The median income of a sample of 300 university professors
B. The mean GPA of all incoming freshman at a state university
Which one of the following is the correct representation of the margin of error when sigma is unknown? A. Z* B. σ/√n C. t*(s/√n) D. x̅±t*(s/√n)
C. t*(s/√n)
Which of the following is NOT a step of hypothesis testing?
Creating an Interval
in order to assess the evidence provided by the sample data, what is the appropriate question to ask?
How likely is it to observe a mean GPA difference of 0.09 or more extreme if there is no difference in the mean GPA for male and female scholarship athletes?
Suppose we have a very right skewed population distribution where μ = 80 and σ = 20. For random samples of size n = 100, what is the mean of the sampling distribution of x̄?
Equal to 80
True or False: Increasing the sample size will lead to a wider margin of error
False
True or False: To be able to safely compute a confidence interval, the only condition that must be met is that the data came from a random sample.
False
True or False: When sigma is unknown it is impossible to compute a confidence interval for μ.
False
True or False: When the value of the population standard deviation is unknown, the only change that occurs in the calculation of our confidence interval is we use s rather than σ.
False
True or False: p-value gives the probability that the null hypothesis is true.
False
Hemoglobin is a protein in the red blood cells that carries oxygen from the lungs to body tissues. People with less than 12 grams of hemoglobin per deciliter of blood (g/dl) are anemic. A public health official in Jordan suspects that the mean hemoglobin level for all children is less than 12 grams. He took a random sample of 50 children and found x̄ = 11.5 g/dl and s = 1.6 g/dl. What are the appropriate null and alternative hypotheses?
H0: μ = 12 vs. Ha: μ < 12
A certain prescription medicine is supposed to contain an average of 250 parts per million (ppm) of a certain chemical. If the concentration is higher than this, the drug may cause harmful side effects; if it is lower, the drug may be ineffective. The manufacturer runs a check to see if the mean concentration in a large shipment conforms to the target level of 250 ppm or not. A simple random sample of 100 portions is tested, and the sample mean concentration is found to be 247 ppm. The sample concentration standard deviation is s = 12 ppm. What are the appropriate null and alternative hypotheses?
H0: μ = 250 vs. Ha: μ ≠ 250
The Federal Pell Grant Program provides need-based grants to low-income undergraduate and certain postbaccalaureate students to promote access to postsecondary education. According to the National Postsecondary Student Aid Study conducted by the U.S. Department of Education in 2008, the average Pell grant award for 2007-2008 was $2,600. Assume that the standard deviation, σ, in Pell grant awards was $500 and that the distribution of awards is left skewed. Suppose we take a random sample of size 20. What is the probability that the mean of the 20 awards is greater than $2,700?
It is not appropriate to calculate probabilities in this situation.
What is the advantage of reporting the average of several measurements rather than the result of a single measurement?
The average of several measurements is more likely to be close to the true mean than the result of a single measurement.
Referring to question 4, what statistic is used to estimate the parameter of interest?
The mean level of nitrogen oxide of a sample of cars of a particular model in the very large fleet.
An administrator in a very large company wants to estimate the mean level of nitrogen oxides (NOX) emitted in the exhaust of a particular car model in their very large fleet of cars. Historically, nitrogen oxide levels have been known to be Normally distributed with a standard deviation of 0.15 g/ml. What is the parameter of interest that the administrator wants to estimate?
The mean level of nitrogen oxide of all cars of a particular model in the very large fleet.
A manufacturing process produces bags of cookies. The weights of these bags are known to be Normally distributed and should have a mean of μ = 15.0 ounces with a standard deviation of σ = 0.4 ounces. In order to monitor the process, four bags are selected periodically and their average weight (x̄) is computed. What is the parameter of interest?
The mean weight of all bags of cookies produced by this manufacturing process.
If all possible samples of size 80 are taken from a population instead of size 20, how would this change the mean and standard deviation of the sampling distribution of x̄?
The mean would stay the same and the standard deviation would decrease
What is the purpose of a statistical control chart?
To distinguish between natural and unnatural variation
True or False: The symbol for the sample standard deviation is "s".
True
if the p-value is less than α, then the results are statistically significant.
True
Suppose you work for an insurance company which covers these types of routine doctor visits for some of its policy holders. The company desires to minimize costs by not sending a healthy patient to a doctor. Which error probability would you choose to make smaller?
Type I error probability
Two researchers used the same sample data to investigate the impact of an energy awareness campaign on the mean monthly energy consumption per household of a large city. Researcher A did a lower-tailed test and Researcher B did a two-sided test. The p-value corresponding to Researcher A's test was found to be 0.030. What would be the correct conclusion to make for Researcher B's test at α = 0.05?
p-value is greater than α = 0.05, therefore, we fail to reject the null hypothesis
What do we use to estimate μ?
x̄
What is the symbol for the sample mean?
x̄