Statistics test 3
The recommended daily allowance of iron for females aged 19-50 is 18 mg/day. A dietitian believes that elderly women (on average) get less than 18 mg/day. The dietitian uses hypothesis testing to check this belief. In this scenario, a Type I error would be: (a) Deciding that elderly women get less than the recommended allowance when they don't. (b) Deciding that elderly women get at least the recommended allowance when they don't.
(a) Deciding that elderly women get less than the recommended allowance when they don't. (You are rejecting Ho when it is true, he thinks that Ha is right which is that they get less when really they get enough)
The average yield per acre for all types of corn in a recent year was 161.9 bushels. An economist believes that the average yield per acre is lower this year. In this scenario, a Type I error would be: (a) Deciding that the yield is lower when it isn't. (b) Deciding that the yield is not lower when it is.
(a) Deciding that the yield is lower when it isn't.
Which of the following statements are true in hypotheses testing?(i) If we reject 𝐻𝑜 when 𝐻𝑜 is in fact true, we made Type I error. (ii) If we reject 𝐻𝑜 when 𝐻𝑜 is in fact false, we made Type II error. (iii) One will reject 𝐻𝑜 if the P-value is smaller than the significance level.
(i) and (iii)
Which of these is not a p-value? 0.7666666666 0 1.001 0.000000000001
1.001
A forensic anthropologist claims that 80% of female skeletons have a sub-pubic angle less than the often-cited 90 degrees. She randomly selects 200 skeletons and finds that 150 have an angle less than 90 degrees. Use 𝛼=0.10α=0.10to test the claim. Suppose the test statistic is -2.1. Which function in Excel finds the p-value?
2*NORM.DIST(-2.1, 0, 1, TRUE)
A fashion company releases a new line of prom dresses that are all under $100. They claim this is a bargain because the average cost of a prom dress is over $300. A customer decides to test this, and takes a random sample of 25 dresses. She finds the mean is $250 with a standard deviation of $10. Assume the population is normally distributed and use 𝛼=0.05 to test the claim. Suppose the test statistic is -1.02. Which function in Excel finds the p-value?
= 1 - T.DIST(-1.02, 24, TRUE)
A veterinarian reads that 15% of dogs are allergic to chicken. However, he claims that the actual proportion is less than this. He randomly selects 15 of his customers and asks them if their dog is allergic to chicken. 3 of them say yes. Assume the population is normally distributed and use 𝛼=0.01 Which function in Excel finds the test statistic? (Note: 3/15 = 0.20)
=(0.20-0.15)/SQRT(0.15*0.85/15)
A nutritionist claims that the average amount of sugar in a 16 oz soda is at least 50 g. He randomly samples 10 sodas and finds they contain an average of 54 g of sugar with a standard deviation of 3 g. Assume the population is normally distributed and use 𝛼=0.10 to test the claim. Which function in Excel finds the test statistic?
=(54-50)/(3/SQRT(10))
A diabetic claims that the average cost of insulin per year for a Type 1 diabetic is $5,705. She takes a sample of 100 Type 1 diabetics and finds their average cost is $5, 912 with a standard deviation of $300. Use 𝛼=0.05 to test the claim. Which function in Excel finds the test statistic?
=(5705-5912)/(300/SQRT(100))
A diabetic claims that the average cost of insulin per year for a Type 1 diabetic is $5,705. She takes a sample of 100 Type 1 diabetics and finds their average cost is $5,912 with a standard deviation of $300. Use 𝛼=0.05 to test the claim. Which function in Excel finds the test statistic?
=(5705-5912)/(300/SQRT(100))
An English professor is studying the use of semicolons over time. She estimates that in the Georgian era, authors used more than 8 semicolons per page. It is well known in her field that the standard deviation of semicolons in this era is 2. She randomly selects 25 pages from different books and finds the average amount of semicolons is 8.5. Assume the population is normally distributed and use 𝛼=0.05 to test the claim. Which formula in Excel finds the test statistic?
=(8.5-8)/(2/SQRT(25))
An English professor is studying the use of semicolons over time. She estimates that in the Georgian era, authors used more than 8 semicolons per page. It is well known in her field that the standard deviation of semicolons in this era is 2. She randomly selects 25 pages from different books and finds the average amount of semicolons is 8.5. Assume the population is normally distributed and use 𝛼=0.05 to test the claim. Also suppose that the test statistic equals 0.85. Which formula in Excel finds the p-value?
=1-NORM.DIST(0.85, 0, 1, TRUE)
A sociologist wishes to test 𝐻0:𝜇=42 vs. 𝐻𝑎:𝜇>42. The sociologist takes a sample of size 100 and calculates a standardized test statistic of 2.34. To calculate a p-value for the test in Excel, the sociologist should use:
=1-NORM.DIST(2.34, 0, 1, TRUE)
A sociologist wishes to test 𝐻0:𝜇=42 vs. 𝐻𝑎:𝜇≠42. The sociologist takes a sample of size 10 and calculates a standardized test statistic of -2.34. To calculate a p-value for the test in Excel, the sociologist should use:
=2*(T.DIST(-2.34, 9, TRUE))
A diabetic claims that the average cost of insulin per year for a Type 1 diabetic is $5,705. She takes a sample of 100 Type 1 diabetics and finds their average cost is $5, 912 with a standard deviation of $300. Use 𝛼=0.05 to test the claim. Suppose the test statistic is 1.65. Which function in Excel finds the p-value?
=2*NORM.DIST(-1.65, 0, 1, TRUE)
A veterinarian reads that 15% of dogs are allergic to chicken. However, he claims that the actual proportion is less than this. He randomly selects 15 of his customers and asks them if their dog is allergic to chicken. 3 of them say yes. Assume the population is normally distributed and use 𝛼=0.01. Suppose the test statistic is -1.11. Which function in Excel finds the p-value?
=NORM.DIST(-1.11, 0, 1, TRUE)
A sociologist wishes to test 𝐻0:𝜇=42 vs. 𝐻𝑎:𝜇<42. The sociologist takes a sample of size 100 and calculates a standardized test statistic of -2.34. To calculate a p-value for the test in Excel, the sociologist should use:
=NORM.DIST(-2.34, 0, 1, TRUE)
A sociologist wishes to test 𝐻0:𝜇=42 vs. 𝐻𝑎:𝜇<42. The sociologist takes a sample of size 10 and calculates a standardized test statistic of -2.34. To calculate a p-value for the test in Excel, the sociologist should use:
=T.DIST(-2.34, 9, TRUE)
A forensic anthropologist claims that 80% of female skeletons have a sub-pubic angle less than the often-cited 90 degrees. She randomly selects 200 skeletons and finds that 150 have an angle less than 90 degrees. Use 𝛼=0.10 to test the claim. Suppose the p-value is 0.25. What is the decision?
Fail to reject Ho
Let p be the proportion of all adults who are in favor of outlawing cigarettes. A researcher testing the hypotheses: 𝐻0:𝑝=0.23;𝐻𝑎:𝑝≠0.23 calculates a p-value of 0.489. The researcher will likely
Fail to reject the null hypothesis.
The alternative hypothesis for a problem is that the population mean is not 19. What is the null hypothesis?
H0 : 𝜇=19
The null hypothesis for a problem is that the population proportion is at least 0.60. What is the alternative hypothesis?
Ha: 𝑝<0.60
On average, people who buy a new car keep that car for 5.3 years. A major automobile manufacturer believes that people who buy a new truck keep it longer. The manufacturer decides to conduct a hypothesis test with a 10% level of significance. They take a sample of 40 people who purchased a new truck, and find that they kept their trucks an average of 7.1 years with a standard deviation of 3.0 years. Set up the null and alternative hypotheses to test whether new truck owners keep their new vehicle for longer (on average) than new car owners do. (a) 𝑥¯b) 𝜇 (c) 𝑝 (d) 𝑝̂ (e) =(f) >(g) <(h) ≠ (i) 10(j) 1.8 (k) 5.3 (m) 7.1
Ho= 𝜇=5.3 Ha=𝜇 >5.3
The null hypothesis for a problem is that the population proportion is at least 0.22. Which type of test will be used?
Left-tailed test
A nutritionist claims that the average amount of sugar in a 16 oz soda is at least 50 g. He randomly samples 10 sodas and finds they contain an average of 54 g of sugar with a standard deviation of 3 g. Assume the population is normally distributed and use 𝛼=0.10 to test the claim. Suppose the p-value is 0.0854. What is the decision?
Reject Ho
Let 𝜇 be the average yield of a genetically modified tomato plant. A farmer tests the following hypotheses: 𝐻0:𝜇=20;𝐻𝑎:𝜇>20, and calculates a p-value of 0.00007. The farmer will likely
Reject the null hypothesis.
If H0: 𝜇≤100, then the test will be _____________.
Right-tailed
A certain type of fiber optic cable transmits light a mean distance of 58 km. A research team wishes to investigate if a modification in the manufacturing process will increase the mean acceptable transmission distance. A sample of 20 batches of cable produced under the new process are tested. The sample mean is 60.3 km with sample standard deviation 2.31 km. Assume the population is normally distributed. To test 𝐻0:𝜇=58 versus 𝐻𝑎:𝜇>58 , the team should use a:
T-test
A certain type of fiber optic cable transmits light a mean distance of 58 km. A research team wishes to investigate if a modification in the manufacturing process will increase the mean acceptable transmission distance. A sample of 20 batches of cable produced under the new process are tested. The sample mean is 60.3 km with sample standard deviation 2.31 km. Assume the population is normally distributed. To test 𝐻0:𝜇=58 versus 𝐻𝑎:𝜇>58, the team should use a:
T-test
A diabetic claims that the average cost of insulin per year for a Type 1 diabetic is $5,705. She takes a sample of 100 Type 1 diabetics and finds their average cost is $5, 912 with a standard deviation of $300. Use 𝛼=0.05 to test the claim. If the decision is to reject 𝐻0, what should the interpretation be?
There is enough evidence to reject the claim
A fashion company releases a new line of prom dresses that are all under $100. They claim this is a bargain because the average cost of a prom dress is over $300. A customer decides to test this, and takes a random sample of 25 dresses. She finds the mean is $250 with a standard deviation of $10. Assume the population is normally distributed and use 𝛼=0.05 to test the claim. Suppose the decision is to reject 𝐻0. Then the interpretation would be:
There is enough evidence to support the claim
If the claim is that the population mean is less than or equal to 2,020 and the decision is to fail to reject H0, what should the interpretation be?
There is not enough evidence to reject the claim.
If the claim is that the population proportion is less than 0.46 and the decision is fail to reject 𝐻0, what should the interpretation be?
There is not enough evidence to support the claim.
If the claim is that the population proportion is less than 0.46 and the decision is fail to reject 𝐻0H0, what should the interpretation be?
There is not enough evidence to support the claim.
If 𝛼>p-value, we reject 𝐻0 True False
True
A standard painkiller is known to bring relief in 3.5 minutes on average (𝜇μ). A new painkiller is hypothesized to bring faster relief to patients. A sample of 40 patients are given the new painkillers. The sample yields a mean of 2.8 minutes and a standard deviation of 1.1 minutes. To test 𝐻0:𝜇=3.5 vs. 𝐻𝑎:𝜇<3.5, the researchers should use a:
Z-test
The null and alternative hypotheses are statements about:
a population parameter
The recommended daily allowance of iron for females aged 19-50 is 18 mg/day. A dietitian believes that elderly women (on average) get less than 18 mg/day. The dietitian uses hypothesis testing to check this belief. In this scenario, a Type 2 error would be: (a) Deciding that elderly women get less than the recommended allowance when they don't. (b) Deciding that elderly women get at least the recommended allowance when they don't.
b
A diabetic claims that the average cost of insulin per year for a Type 1 diabetic is $5,705. She takes a sample of 100 Type 1 diabetics and finds their average cost is $5, 912 with a standard deviation of $300. Use 𝛼=0.05 to test the claim. If the decision is to reject 𝐻0, what should the interpretation be? There is
enough evidence to reject the claim
A researcher wishes to test the following hypotheses: 𝐻0:𝑝=0.23;𝐻𝑎:𝑝≠0.23. The researcher takes a sample and performs a test which yields a p-value of 0.092. At a significance level of 5%, the researcher should _________ the null hypothesis. There _______ sufficient evidence to claim that the proportion differs from 23%.
fail to reject, is not
The average room rate in hotels in a certain region is $82.53. A travel agent believes that the average in a particular resort area is different. The agent tests 𝐻0:𝜇=82.53; 𝐻𝑎:𝜇≠82.53, and calculates a p-value of 0.063. At a 5% level of significance, the agent should fail to reject the null hypothesis. There is not sufficient evidence to claim that the average in the resort area differs from that of the region.
fail to reject, there is not enough evidence to support the claim
Steps for p-value method
state hypotheses find test stat find p value make decision provide interpretation
A fashion company releases a new line of prom dresses that are all under $100. They claim this is a bargain because the average cost of a prom dress is over $300. A customer decides to test this, and takes a random sample of 25 dresses. She finds the mean is $250 with a standard deviation of $10. Assume the population is normally distributed and use 𝛼=0.05 to test the claim. Which distribution should be used to test the claim?
t-distribution
A nutritionist claims that the average amount of sugar in a 16 oz soda is at least 50 g. He randomly samples 10 sodas and finds they contain an average of 54 g of sugar with a standard deviation of 3 g. Assume the population is normally distributed and use α=0.10to test the claim. Which distribution should be used?
t-distribution
Graphically, the p-value is ______________.
the area of the tail of the distribution bounded by the test statistic.
Let 𝜇 be the average starting salary for graduates with a master's degree in business. A university administrator tests the following hypotheses: 𝐻0:𝜇=83000;𝐻𝑎:𝜇<83000, and calculates a p-value of 0.004. At the 5% level of significance, the administrator should conclude that:
the average starting salary is less than $83000
The p-value is _______________.
the probability of obtaining a value as extreme or more extreme than the observed value by chance alone, assuming the null hypothesis is true.
If 𝐻0: 𝜇=969, then the test will be _____________.
two-tailed
A university administrator wishes to know if there is a difference in average starting salary for graduates with master's degrees in engineering and those with master's degrees in business. The average starting salary for graduates with master's degrees in engineering $83,000. The administrator samples 32 former students with master's degrees in business and finds an average starting salary of $61,000. The administrator tests the following hypotheses: 𝐻0:𝜇=83000;𝐻𝑎:𝜇<83000 In this scenario, x¯= and μ0=
xbar= 61000 μ0=83000
An English professor is studying the use of semicolons over time. She estimates that in the Georgian era, authors used more than 8 semicolons per page. It is well known in her field that the standard deviation of semicolons in this era is 2. She randomly selects 25 pages from different books and finds the average amount of semicolons is 8.5. Assume the population is normally distributed and use 𝛼=0.05 to test the claim. Which distribution should be used?
z-distribution
A veterinarian reads that 15% of dogs are allergic to chicken. However, he claims that the actual proportion is less than this. He randomly selects 15 of his customers and asks them if their dog is allergic to chicken. 3 of them say yes. Assume the population is normally distributed and use 𝛼=0.01. State the hypotheses.
𝐻0:𝑝≥0.15 𝐻𝑎:𝑝<0.15(𝑐𝑙𝑎𝑖𝑚)
A fashion company releases a new line of prom dresses that are all under $100. They claim this is a bargain because the average cost of a prom dress is over $300. A customer decides to test this, and takes a random sample of 25 dresses. She finds the mean is $250 with a standard deviation of $10. Assume the population is normally distributed and use 𝛼=0.05 to test the claim. State the hypotheses.
𝐻0:𝜇≤300 𝐻𝑎:𝜇>300 (𝑐𝑙𝑎𝑖𝑚)
In a random sample of 200 adults, 54 say they are in favor of outlawing cigarettes. Let p be the proportion of all adults who are in favor of outlawing cigarettes. A researcher wishes to test the following hypotheses: 𝐻0:𝑝=0.23;𝐻𝑎:𝑝≠0.23. In this scenario, the appropriate test statistic is:
𝑍=𝑝̂ −𝑝𝑜 ____________ 𝑝𝑜𝑞𝑜/√n
A university administrator wishes to know if there is a difference in average starting salary for graduates with master's degrees in engineering and those with master's degrees in business. The average starting salary for graduates with master's degrees in engineering $83,000. The administrator samples 42 former students with master's degrees in business and finds an average starting salary of $61,000. The administrator tests the following hypotheses: 𝐻0:𝜇=83000;𝐻𝑎:𝜇<83000. In this scenario, the appropriate test statistic is:
𝑍=𝑥¯−𝜇0 ______________ 𝑠/√n
In a random sample of 200 adults, 54 say they are in favor of outlawing cigarettes. Let p be the proportion of all adults who are in favor of outlawing cigarettes. A researcher wishes to test the following hypotheses: 𝐻0:𝑝=0.23;𝐻𝑎:𝑝≠0.23 In this scenario,
𝑝̂ =0.27 po=0.23
If 𝐻0: 𝜇≤100, then 𝐻a:
𝜇>100