Download Review for Test 5 STA 2023 spr 2014

Review for Test 5 STA 2023
spr 2014
Name___________________________________
Given the linear correlation coefficient r and the sample size n, determine the critical values of r and use your finding to
state whether or not the given r represents a significant linear correlation. Use a significance level of 0.05.
1) r = 0.523, n = 25
Answer: Critical values: r = ±0.396, significant linear correlation
Construct a scatterplot for the given data.
x 1 -7 -2 -4 1 5 -6 7 -4 -2
2)
y 7 -1 -7 5 2 3 -2 1 -5 -3
Answer:
Determine which scatterplot shows the strongest linear correlation.
3) Which shows the strongest linear correlation?
A)
B)
Answer: A
1
C)
Find the value of the linear correlation coefficient r.
4) x 23.6 35.3 18.3 49.0 13.2
y
4
9
3
3
8
Answer: -0.2
5) x 57 53 59 61 53 56 60
y 156 164 163 177 159 175 151
Answer: 0.109
6) The paired data below consist of the temperatures on randomly chosen days and the amount a certain kind of
plant grew (in millimeters):
Temp 62 76 50 51 71 46 51 44 79
Growth 36 39 50 13 33 33 17 6 16
Answer: 0.196
7) Two separate tests are designed to measure a student's ability to solve problems. Several students are randomly
selected to take both tests and the results are shown below.
Test A 48 52 58 44 43 43 40 51 59
Test B 73 67 73 59 58 56 58 64 74
Answer: 0.867
Suppose you will perform a test to determine whether there is sufficient evidence to support a claim of a linear
correlation between two variables. Find the critical values of r given the number of pairs of data n and the significance
level .
8) n = 40, = 0.01
Answer: r = ±0.402
9) n = 14,
= 0.05
Answer: r = ±0.532
Use the given data to find the best predicted value of the response variable.
^
10) Four pairs of data yield r = 0.942 and the regression equation y = 3x. Also, y = 12.75. What is the best predicted
value of y for x = 4.3?
Answer: 12.75
^
11) Eight pairs of data yield r = 0.708 and the regression equation y = 55.8 + 2.79x. Also, y = 71.125. What is the best
predicted value of y for x = 6.8?
Answer: 74.77
12) The regression equation relating dexterity scores (x) and productivity scores (y) for the employees of a company
^
is y = 5.50 + 1.91x. Ten pairs of data were used to obtain the equation. The same data yield r = 0.986 and y = 56.3.
What is the best predicted productivity score for a person whose dexterity score is 32?
Answer: 66.62
^
13) Six pairs of data yield r = 0.789 and the regression equation y = 4x - 2. Also, y = 19.0. What is the best predicted
value of y for x = 5?
Answer: 19.0
2
^
14) Ten pairs of data yield r = 0.003 and the regression equation y = 2 + 3x. Also, y = 5.0. What is the best predicted
value of y for x = 2?
Answer: 5.0
Use the given data to find the equation of the regression line. Round the final values to three significant digits, if
necessary.
15) x 2 4 5 6
y 7 11 13 20
^
Answer: y = 3.0x
16) x 0 3 4 5 12
y 8 2 6 9 12
^
Answer: y = 4.88 + 0.525x
17) x 24 26 28 30 32
y 15 13 20 16 24
^
Answer: y = -11.8 + 1.05x
18) Managers rate employees according to job performance and attitude. The results for several randomly selected
employees are given below.
Performance 59 63 65 69 58 77 76 69 70 64
Attitude
72 67 78 82 75 87 92 83 87 78
^
Answer: y = 11.7 + 1.02x
Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final
conclusion that addresses the original claim.
19) A manufacturer considers his production process to be out of control when defects exceed 3%. In a random
sample of 85 items, the defect rate is 5.9% but the manager claims that this is only a sample fluctuation and
production is not really out of control. At the 0.01 level of significance, test the manager's claim.
Answer: H0 : p = 0.03. H1 : p > 0.03. Test statistic: z = 1.57. P-value: p = 0.0582.
Critical value: z = 2.33. Fail to reject null hypothesis. There is not sufficient evidence to warrant rejection
of the manager's claim that production is not really out of control.
20) A supplier of digital memory cards claims that no more than 1% of the cards are defective. In a random sample
of 600 memory cards, it is found that 3% are defective, but the supplier claims that this is only a sample
fluctuation. At the 0.01 level of significance, test the supplier's claim that no more than 1% are defective.
Answer: H0 : p = 0.01. H1 : p > 0.01. Test statistic: z = 4.92. P-value: p = 0.0001.
Critical value: z = 2.33. Reject null hypothesis. There is sufficient evidence to warrant rejection of the
claim that no more than 1% are defective. Note: Since the term "no more than" is used, the translation is p
0.01. Therefore, the competing hypothesis is p > 0.01.
Find the P-value for the indicated hypothesis test.
21) A medical school claims that more than 28% of its students plan to go into general practice. It is found that
among a random sample of 130 of the school's students, 32% of them plan to go into general practice. Find the
P-value for a test of the school's claim.
Answer: 0.1539
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22) In a sample of 88 children selected randomly from one town, it is found that 8 of them suffer from asthma. Find
the P-value for a test of the claim that the proportion of all children in the town who suffer from asthma is equal
to 11%.
Answer: 0.5686
Solve the problem.
23) A manufacturer finds that in a random sample of 100 of its CD players, 96% have no defects. The manufacturer
wishes to make a claim about the percentage of nondefective CD players and is prepared to exaggerate. What is
the highest rate of nondefective CD players that the manufacturer could claim under the following condition?
His claim would not be rejected at the 0.05 significance level if this sample data were used. Assume that a
left-tailed hypothesis test would be performed.
Answer: 98.2%
Determine whether the given conditions justify testing a claim about a population mean µ.
24) The sample size is n = 21, = 6.07, and the original population is normally distributed.
Answer: Yes
25) The sample size is n = 19,
is not known, and the original population is normally distributed.
Answer: Yes
Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final
conclusion that addresses the original claim.
26) Various temperature measurements are recorded at different times for a particular city. The mean of 20°C is
obtained for 60 temperatures on 60 different days. Assuming that = 1.5°C, test the claim that the population
mean is 22°C. Use a 0.05 significance level.
Answer: H0 : µ = 22; H1: µ 22. Test statistic: z = -10.33. P-value: 0.0002. Because the P-value is less than the
significance level of = 0.05, we reject the null hypothesis. There is sufficient evidence to warrant
rejection of the claim that the population mean temperature is 22°C.
Test the given claim. Use the P-value method or the traditional method as indicated. Identify the null hypothesis,
alternative hypothesis, test statistic, critical value(s) or P-value, conclusion about the null hypothesis, and final
conclusion that addresses the original claim.
27) A simple random sample of 15-year old boys from one city is obtained and their weights (in pounds) are listed
below. Use a 0.01 significance level to test the claim that these sample weights come from a population with a
mean equal to 149 lb. Assume that the standard deviation of the weights of all 15-year old boys in the city is
known to be 16.2 lb. Use the traditional method of testing hypotheses.
147 138 162 151 134 189 157 144 175 127 164
Answer: H0 : µ = 149 lb
H1 : µ 149 lb
Test statistic: z = 0.91
Critical-values: z = ± 2.575
Do not reject H0 ; At the 1% significance level, there is not sufficient evidence to warrant rejection of the
claim that these sample weights come from a population with a mean equal to 149 lb.
Determine whether the hypothesis test involves a sampling distribution of means that is a normal distribution, Student t
distribution, or neither.
28) Claim: µ = 981. Sample data: n = 24, x = 972, s = 26. The sample data appear to come from a normally
distributed population with = 28.
Answer: Normal
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29) Claim: µ = 120. Sample data: n = 11, x = 100, s = 15.2. The sample data appear to come from a normally
distributed population with unknown µ and .
Answer: Student t
Assume that a simple random sample has been selected from a normally distributed population. Find the test statistic,
P-value, critical value(s), and state the final conclusion.
30) Test the claim that for the population of female college students, the mean weight is given by µ = 132 lb. Sample
data are summarized as n = 20, x = 137 lb, and s = 14.2 lb. Use a significance level of
Answer:
= 0.1.
= 0.1
Test statistic: t = 1.57
P-value: p = 0.1318
Critical values: t = ±1.729
Because the test statistic, t < 1.729, we fail to reject the null hypothesis. There is not sufficient evidence to
warrant rejection of the claim that µ = 132 lb.
31) Test the claim that for the adult population of one town, the mean annual salary is given by µ = $30,000. Sample
data are summarized as n = 17, x = $22,298, and s = $14,200. Use a significance level of
Answer:
= 0.05.
= 0.05
Test statistic: t = -2.236
P-value: p = 0.0399
Critical values: t = ±2.120
Because the test statistic, t < -2.120, we reject the null hypothesis. There is sufficient evidence to warrant
rejection of the claim that µ = $30,000.
Assume that a simple random sample has been selected from a normally distributed population and test the given claim.
Use either the traditional method or P-value method as indicated. Identify the null and alternative hypotheses, test
statistic, critical value(s) or P-value (or range of P-values) as appropriate, and state the final conclusion that addresses the
original claim.
32) Use a significance level of = 0.05 to test the claim that µ = 32.6. The sample data consist of 15 scores for which
x = 39 and s = 7.8. Use the traditional method of testing hypotheses.
Answer: H0 : µ = 32.6. H1 : µ 32.6. Test statistic: t = 3.18. Critical values: t = ±2.145. Reject H0 . There is sufficient
evidence to warrant rejection of the claim that the mean is 32.6.
33) In tests of a computer component, it is found that the mean time between failures is 520 hours. A modification is
made which is supposed to increase the time between failures. Tests on a random sample of 10 modified
components resulted in the following times (in hours) between failures.
518 548 561 523 536
499 538 557 528 563
At the 0.05 significance level, test the claim that for the modified components, the mean time between failures is
greater than 520 hours. Use the P-value method of testing hypotheses.
Answer: H0 : µ = 520 hrs. H1: µ > 520 hrs. Test statistic: t = 2.612.
0.01 < P-value < 0.025. Reject H0. There is sufficient evidence to support the claim that the mean is
greater than 520 hours.
34) Use a significance level of = 0.01 to test the claim that µ > 2.85. The sample data consist of 9 scores for which x
= 3.1 and s = 0.6. Use the traditional method of testing hypotheses.
Answer: H0 : µ = 2.85. H1: µ > 2.85. Test statistic: t = 1.25. Critical value: t = 2.896. Fail to reject H0 . There is not
sufficient evidence to support the claim that the mean is greater than 2.85.
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Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.
35) n = 51, x = 27; 95% confidence
Answer: 0.392 < p < 0.666
36) Of 234 employees selected randomly from one company, 12.82% of them commute by carpooling. Construct a
90% confidence interval for the true percentage of all employees of the company who carpool.
Answer: 9.23% < p < 16.4%
Solve the problem.
37) In a certain population, body weights are normally distributed with a mean of 152 pounds and a standard
deviation of 26 pounds. How many people must be surveyed if we want to estimate the percentage who weigh
more than 180 pounds? Assume that we want 96% confidence that the error is no more than 4 percentage points.
Answer: 658
Use the confidence level and sample data to find a confidence interval for estimating the population µ. Round your
answer to the same number of decimal places as the sample mean.
38) Test scores: n = 92, x = 90.6,
= 8.9; 99% confidence
Answer: 88.2 < µ < 93.0
39) A random sample of 130 full-grown lobsters had a mean weight of 21 ounces and a standard deviation of 3.0
ounces. Construct a 98% confidence interval for the population mean µ.
Answer: 20 oz < µ < 22 oz
Use the given information to find the minimum sample size required to estimate an unknown population mean µ.
40) How many women must be randomly selected to estimate the mean weight of women in one age group. We
want 90% confidence that the sample mean is within 3.7 lb of the population mean, and the population standard
deviation is known to be 28 lb.
Answer: 155
Use the given degree of confidence and sample data to construct a confidence interval for the population mean µ. Assume
that the population has a normal distribution.
41) A laboratory tested twelve chicken eggs and found that the mean amount of cholesterol was 225 milligrams
with s = 15.7 milligrams. Construct a 95% confidence interval for the true mean cholesterol content of all such
eggs.
Answer: 215.0 mg < µ < 235.0 mg
42) The football coach randomly selected ten players and timed how long each player took to perform a certain
drill. The times (in minutes) were:
7.2 10.5 9.9 8.2 11.0
7.3 6.7 11.0 10.8 12.4
Determine a 95% confidence interval for the mean time for all players.
Answer: 8.15 min < µ < 10.85 min
Find the indicated probability.
43) A die with 6 sides is rolled. What is the probability of rolling a number less than 5?
Answer:
2
3
6