Skip to main content
Statistics LibreTexts

Answers to most problems

  • Page ID
    5450
  • Answers are provided for most problems so you can immediately check your answers to see if you are doing it correctly. This should facilitate learning. Answers are not provided for some problems to simulate real-world conditions and tests, since answers are not known in either case.

    Chapter 1

    1a. parameter

    1b. statistic

    2. parameter

    4a. \(H_0: \mu = 20\) \(H_1: \mu > 20\)

    4c. \(H_0: \mu_A = \mu_C\) \(H_1: \mu_A \ne \mu_C\)

    4d. \(H_0: p_m = p_A\) \(H_1: p_m \ne p_A\)

    6.

    p-value \(\alpha\) Hypothesis \(H_0\) or \(H_1\) Significant or Not Significant Error
    Type I or Type II
    0.043 0.05 \(H_1\) Significant Type I
    0.32 0.05 \(H_0\) Not Significant Type II
    \(5.6 \times 10^{-6}\) 0.05 \(H_1\) Significant Type I
    7.3256 0.01 x x x

    7a. At the 5% level of significance, the proportion is significantly greater than 0.5 (p = 0.022, n = 350).

    7b. At the 1% level of significance, the proportion is not significantly less than 0.25 (p = 0.048, n = 1400).

    7d. At the 5% level of significance, the mean is different than 20 (\(5.6 \times 10^{-5}\), n = 32).

    8.

    屏幕快照 2019-05-22 下午3.43.27.png

    8a. Left

    8c. 0.40

    8e. 0.66

    8h. 0.155

    8i. \(H_0\)

    8j. No

    8k. type II

    8l. At the 2% level of significance, the proportion is not significantly less than 0.5 (p = 0.155, n = 80).

    9.

    屏幕快照 2019-05-22 下午3.44.47.png

    9a. Right

    9c. 340

    9d. 0.035

    9f. 0.67

    9h. 0.005

    9i. H1

    9j. Yes

    9l. At the 0.035 level of significance, the mean is significantly greater than 300 (p = 0.005, n = 10).

    10a. \(H_0: p = 0.5\) \(H_1: p > 0.5\)

    10b. Right

    屏幕快照 2019-05-22 下午3.46.28.png

    10d. 60

    10f. 0.56

    10g. 0.44

    10h. 0

    10i. At the 2% level of significance, the proportion who survive cancer at least 5 years is significantly greater than 0.5 (p = 0, n = 100).

    Chapter 2

    1.

    Research Design Table
    Research Question: which route has a faster average time
    Type of Research Observational Study
    Observational Experiment
    Manipulative Experiment
    What is the response variable? Time it takes for the commute
    What is the parameter that will be calculated? Mean Proportion Correlation
    List potential latent variables Think of at least 2 yourself

    Grouping/explanatory Variables 1 (if present)

    routes

    Levels:

    Route 1 and Route 2

    3.

    Research Design Table
    Research Question: Which is more effective at increasing biodiversity, the hands-off approach or the deliberate approach?
    Type of Research Observational Study
    Observational Experiment
    Manipulative Experiment
    What is the response variable? Number of species
    What is the parameter that will be calculated? Mean Proportion Correlation
    List potential latent variables Think of at least 2 yourself

    Grouping/explanatory Variables 1 (if present)

    approaches

    Levels:

    hands-off deliberate control

    4b.

    Research Design Table
    Research Question: Does static or dynamic stretching result in improvement in flexibility in the largest proportion of people?
    Type of Research Observational Study
    Observational Experiment
    Manipulative Experiment
    What is the response variable? Improvement in sit and reach test
    What is the parameter that will be calculated? Mean Proportion
    List potential latent variables Think of at least 2 yourself

    Grouping/explanatory Variables 1 (if present)

    Stretching method

    Levels:

    static dynamic

    8a. 102 N, 40 N, 18 Y, 49 N, 61 N, 60 N, 57 N, 16 N, 90 N, 46 Y,

    135 N, 105 Y, 83 N, 102 N, 3 N, 70 Y, 47 N, 42 N, 5 N, 68 N,

    Sample Proportion __4/20 = 0.2

    8b. West 37 Y,45 N, 21 N, 56 N, 70 Y, 68 N, 65 N, 18 Y, 22 N, 52 Y, 75 Y,

    East 93 N, 105 Y, 109 N, 90 N, 114 N, 137 Y, 133 N, 131 N, 104 Y

    Sample Proportion _8/20 = 0.4

    8c. 2 Y, 9 N, 16 N, 23 N, 30 N, 37 Y, 44 Y, 51 Y, 58 N, 65 N,
    72 Y, 79 N, 86 Y, 93 N, 100 N, 107 N, 114 N, 121 N, 128 N, 135 N,

    Sample Proportion _6/20 = .30

    8d. Which cluster is selected? ___7____ Sample Proportion _9/20=0.45

    8m 屏幕快照 2019-05-22 下午3.57.45.png

    9a.

    Research Design Table
    Research Question: Does raising the minimum wage cause unemployment to increase?
    Type of Research Observational Study
    Observational Experiment
    Manipulative Experiment
    What is the response variable? Change in Unemployment rate
    What is the parameter that will be calculated? Mean Proportion Correlation
    List potential latent variables Think of at least 2 yourself

    Grouping/explanatory Variables 1 (if present)

    State minimum wage change

    Levels:

    Increase minimum wage

    No change in minimum wage

    9b. Cluster

    9c. 2004, 2012, 2006

    9d. Provide your own thoughtful answer.

    9e. Provide your own thoughtful answer.

    9f. Provide your own thoughtful answer.

    9g. At the 5% level of significance, there is not a significant difference in the change in unemployment rate between states that raised their minimum wage and those that didn’t (p = 0.286).

    10a.

    Research Design Table
    Research Question: Will the number of falls increase after bedrails are removed?
    Type of Research Observational Study
    Observational Experiment
    Manipulative Experiment
    What is the response variable? Falls
    What is the parameter that will be calculated? Mean Proportion Correlation
    List potential latent variables Think of at least 2 yourself

    Grouping/explanatory Variables 1 (if present)

    Bedrails

    Levels:

    Present

    Not present

    10b. At the 5% level of significance, there was not a significant increase in the number of falls per 10,000 bed days after the implementation of the new policy (p = 0.18).

    10c. There were fewer serious falls, more minor and no-injury falls. A possible reason is that the falls are from a lower height since the patient isn’t crawling over the top of the rails.

    10d. Provide your own thoughtful response.

    Chapter 3

    1.

    屏幕快照 2019-05-22 下午4.04.11.png

    2.

    屏幕快照 2019-05-22 下午4.04.42.png

    3. mean = 43, Standard deviation = 4.78, variance = 22.89

    4.

    屏幕快照 2019-05-22 下午4.05.48.png

    屏幕快照 2019-05-22 下午4.06.10.png

    5. \(r = \dfrac{\text{cov}(x, y)}{s_x s_y} = \dfrac{2.09}{5.56 \cdot 1.20} = 0.313\)

    6a.

    Research Design Table
    Research Question: Is average number of problems answered correctly in one minute was greater for students who passed the class than for those who didn’t pass?
    Type of Research Observational Study
    Observational Experiment
    Manipulative Experiment
    What is the response variable? Number of correctly answered problems
    What is the parameter that will be calculated? Mean Proportion Correlation
    List potential latent variables

    Grouping/explanatory Variables 1 (if present)

    Success in course

    Levels:


    Pass Fail

    Grouping/explanatory Variables 2 (if present) Levels:


    6b. Calc: 2,9

    6c. Cluster

    6d. Quantitative discrete

    6e.

    屏幕快照 2019-05-22 下午4.10.36.png

    6f. Provide your own thoughtful response.

    6g.

    Mean Variance Standard Deviation
    Failed 15.89 33.88 5.82
    Passed

    6h. At the 5% level of significance, the average automaticity score of those who pass the class is significantly more than the score of those who fail the class (p = 0.0395, \(n_{\text{fail}}\) = 19, \(n_{\text{pass}}\) = 49).

    6i. Yes

    7a.

    Research Design Table
    Research Question: Is the average heart rate lower with alcohol than with water?
    Type of Research Observational Study
    Observational Experiment
    Manipulative Experiment
    What is the response variable? heart rate
    What is the parameter that will be calculated? Mean Proportion Correlation
    List potential latent variables List you own ideas

    Grouping/explanatory Variables 1 (if present)

    drop

    Levels:


    water (first time) alcohol, water (second time)

    Grouping/explanatory Variables 2 (if present) Levels:


    7b.

    屏幕快照 2019-05-22 下午4.14.19.png

    7c.

    Heart Rate after Alcohol Heart Rate after Water
    mean 32.8 52.1
    Standard Deviation 10.7 13.0
    Median 32.5 54.0

    7d. At the 5% level of significance, the average daphnia heart rate with alcohol is significantly less than the average daphnia heart rate with water (p = \(1.28 \times 10^{-5}\), \(n_{\text{alcohol}}\) = 18, \(n_{\text{water}}\) = 18).

    7e. Provide your own thoughtful answer.

    Chapter 4

    1.
    1a. P(S) = 0.70

    1b. P(F) = 0.30

    1c. P(FSSSF) = P(F)P(S) P(S) P(S) P(F) = (0.3) (0.7) (0.7) (0.7) (0.3) = 0.03087

    1d. 10

    1e. 0.3087

    1f.

    屏幕快照 2019-05-22 下午4.17.49.png

    1g. \(\mu = np = 5(0.7) = 3.5\) \(\sigma = \sqrt{npq} = \sqrt{5 (0.7) (0.3)} = 1.025\)

    1h. \(P(X \le 3)\) = binomcdf(n, p, x) = binomcdf (5, 0.70, 3) = 0.4718. The null is supported. The data are not significant.

    2.

    2a. P(S) = 0.40

    2b. P(F) = 0.60

    2c. 0.0036864

    2d. 21

    2e. 0.0774

    2f.

    \(X = x\) 0 1 2 3 4 5 6 7
    \(P(X = x)\) 0.0280 0.1306 0.2613 0.2903 0.1935 0.0774 0.0172 0.0016

    2g. 2.8, 1.296

    2h. P-value = 0.0963 alternative

    3.

    3j. P-value = 0.047

    At the 5% level of significance, the proportion of residents opposed to the terminals is significantly greater than 0.5 (p = 0.047, n = 300)

    3k. z = 1.73 p-value 0.0418

    At the 5% level of significance, the proportion of residents opposed to the terminals is significantly greater than 0.5 (z = 1.73, p = 0.0418, n = 300).

    3l. 0.55

    3m. 0.02887

    3n. z = 1.73 p-value 0.0418

    At the 5% level of significance, the proportion of residents opposed to the terminals is significantly greater than 0.5 (z = 1.73, p = 0.0418, n = 300).

    4a. \(H_0: \mu = 43,362\) and \(H_1: \mu < 43,362\)

    4b. \(\mu = 43,362\)

    4c. \(\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}} = \dfrac{7900}{\sqrt{10}} = 2498\)

    4d.

    屏幕快照 2019-05-22 下午4.25.12.png

    X axis: 35868 38366 40864 43362 45860 48358 50856

    4e. 18,225 (use stat-edit and then stat-calc-1-var stats)

    4f. z = -10.06 p-value < 0.0002

    6b. \(H_0: \mu = 54.1\) and \(H_1: \mu > 54.1\)

    6d. \(bar{x} = 46.73\), \(s = 16.377\)

    6e. \(\mu = 54.1\) \(\sigma_{\bar{x}} = 2.939\)

    6f.

    屏幕快照 2019-05-22 下午4.25.12.png

    45.4 48.3 51.2 54.1 57 59.9 62.8

    6g. z = -2.51 p-value 0.9940

    At the 5% level of significance, the average walk score of small cities is not significantly greater than big cities (z = -2.51, p = 0.994, n = 30)

    7e. p-value = 0.3865 0.0148

    7g p-value = 0.0129

    7h z = 2.229, p= 0.0129

    8c.

    Impact Control
    Mean 455
    Standard Deviation 614.8
    Median 150

    Chapter 5

    1a. \(H_0: \mu_T = \mu_D\) \(H_1: \mu_T \ne \mu_D\) Test: 2 independent samples t test

    1b. \(H_0: p = 0.5\) \(H_1: p > 0.5\) Test: 1 proportion z test

    1c. \(H_0: p_{STEM} = p_{SS}\) \(H_1: p_{STEM} \ne p_{SS}\) Test: 2 proportion Z test

    1e. \(H_0: \mu = 7\) \(H_1: \mu > 7\) Test: 1 sample t test

    1g. \(H_0: \mu = 0\) \(H_1: \mu < 0\) Test: 1 sample t test

    2a. \(H_0: \mu = 15\), \(H_1: \mu < 15\)

    2b. Test the bypothesis:

    \(t = \dfrac{\bar{x} - \mu}{\dfrac{s}{\sqrt{n}}}\) \(t = \dfrac{14.28 - 15}{\dfrac{4.6}{\sqrt{30}}}\) t = -0.857 p > 0.1 or p = 0.1991

    Formula Substitution Test Statistic p-value

    2c. Fill in the blanks for the concluding sentence. At the _5%__ level of significance, the mean money spent per day _is not_ significantly less than $15 (t = ____________, p ___________, df = 29).

    3a. Write the hypotheses: \(H_0: p = 0.5\) , \(H_1: p > 0.5\), Sample proportion \((\hat{p}\)) = \(\dfrac{118}{179}\) = 0.659

    3b. Test the hypothesis:

    \(z = \dfrac{\hat{p} - p}{\sqrt{\dfrac{p(1 - p}{n}}}\) \(z = \dfrac{0.659 - 0.5}{\sqrt{\dfrac{0.5(1 - 0.5}{179}}}\) p < 0.0002 or 1.02 \(\times 10^{-5}\)

    Formula Substitution Test Statistic p-value

    Write the concluding sentence: At the 5% level of significance, the average price on Tuesday is not significantly less than other days (t = -0.479, p > 0.25, n = 7).

    5a. Write the hypotheses: \(H_0:\) _ \(\mu_{40} = \mu_{60}\) ___, \(H_1:\) __\(\mu_{40} < \mu_{60}\)__,

    5c. Write the concluding sentence: At the 10% level of significance, the mean guess at Morocco’s population by people with low phone digits is significantly less than the mean guess of those with high phone digits (\(t\) = -1.835, \(p\) < 0.05, \(n_{40}\) = 15, \(n_{60}\) = 15).

    6c. Test the hypothesis: Test Statistic = -12.41, p-value = p =1

    7e. Write a concluding sentence. At the 5% level of significance, the proportion of \(12^{\text{th}}\) grade students using drugs in 2012 is not significantly greater than in 2002 (\(z\) = 0.819, \(p\) = 0.2061, \(n_{2012}\) = 630, \(n_{2002}\) = 2184).

    8a.

    Research Design Table
    Research Question: Is there a significant difference in the mean times of the men and women who finish the triathlon course?
    Type of Research Observational Study
    Observational Experiment
    Manipulative Experiment
    What is the response variable? heart rate
    What is the parameter that will be calculated? Mean Proportion Correlation
    List potential confounding variables

    Grouping/explanatory Variables 1 (if present)

    Gender

    Levels:


    Men, Women

    8b. Write the hypotheses. \(H_0: \mu_{\text{men}} = \mu_{\text{women}}\), \(H_1: \mu_{\text{men}} \ne \mu_{\text{women}}\)

    9a. Conclusion: At the 5% level of significance, there is not a significant difference in the proportion of days that African Americans and LGB individuals record at least one stigma-related stressor (\(z\) = 0.062, \(p = 0.950\), \(n_{AA}\) = 190, \(n_{LGB}\) = 310)

    9b. Conclusion: At the 5% level of significance, the mean psychological distress score for those using rumination is significantly different than for those using distraction (\(t\) = 2.189, \(p\) = 0.033, \(n_R\) = 26, \(n_D\) = 26)

    Chapter 6

    1. 0.778

    Margin of Error 0.060 Confidence Interval (0.718,0.838) Calculator confidence interval (0.71853,0.83823)

    2. Girls: 0.743, Boys 0.778

    Margin of Error 0.132, Confidence Interval (9-0.167,0.097)

    3. (Note: There are 112 degrees of freedom. This df does not appear in your tables. It falls between 60 df and 120 df. To make sure that the interval is sufficiently large, the critical t value for 60 df will be used. The actual value, as found using the Excel function T.INV.2T(0.1,112) is 1.6586).

    Margin of error 14.6 (73.4,102.6)

    Calculator confidence interval (73.49,102.51)

    5. Point estimate 156

    Margin of error 92.9 Confidence Interval (63.1,248.9)

    Calculator confidence interval (63.087,248.91)

    6. Point Estimate 5.2 degrees

    Margin of error 3.2 confidence interval (2,8.4)

    Calculator confidence interval (2.0338,8.3662)

    8. Point estimate 0.116

    Margin of Error 0.017, Confidence interval (0.099,0.133)

    9.

    Margin of Error 1% 5% 10% 20%
    Sample Size 9604

    10.

    Degree of Confidence 99% 95% 90% 80%
    Sample Size 1844

    11a. Stratified

    11b. 256 , 379 ,

    11d. Mean = 59.5, SD = 23.78

    11e. (48.4, 70.6)

    11f. Lowest: 290.4

    Chapter 7

    屏幕快照 2019-05-23 下午1.13.51.png

    1b Mean batting average 0.2516 Standard deviation for batting average 0.0155

    Mean runs scored 676.1 Standard deviation for runs scored 82.20

    1c. At the 5% level of significance, there is a significant correlation between batting average and runs scored (\(t\) = 3.84, \(p\) < 0.01, \(n\) = 10).

    1d. Regression equation: y = -397.98 + 4269x

    1e. \(r^2\) = 0.6479. It means 64.8% of the total variation from the mean for team runs scored is attributed to the variation in the batting average.

    1f. 669.285

    1g.

    屏幕快照 2019-05-23 下午1.16.39.png

    Correlation: 0.955

    Hypothesis test concluding sentence: At the 5% level of significance, there is a significant correlation between slugging percentage and runs scored (\(t\) = 9.10, \(p\) = 1.699E-5, \(n\) = 10).

    Regression equation: \(y = -408.83 + 2741.80x\)

    Coefficient of determination (\(r^2\)): 0.912

    Predict the number of runs scored for a team with a slugging percentage of 0.400. 687.89

    2a.

    屏幕快照 2019-05-23 下午1.18.34.png

    2b.

    屏幕快照 2019-05-23 下午1.18.56.png

    \(R\) = 0.0359, \(r^2\) = 0.0013, \(y = 3.05 + 0.0067x\)

    At the 5% level of significance, there is not a significant correlation between spending on education and spending on public assistance (\(t\) = -1.35, \(p\) = 0.215, \(n\) = 10).

    3.

    屏幕快照 2019-05-23 下午1.20.04.png

    \(Y = 49075.6 +156.2x\) \(r = 0.097\), \(r^2 = 0.0094\), \(t = 0.258\), \(p = 0.8038\).

    4a.

    屏幕快照 2019-05-23 下午1.21.17.png

    \(Y=-36.38+0.0023x\) \(r = 0.929\), \(r^2 = 0.863\), \(t = 6.64\), \(p = 2.92 \times 10^{-4}\)

    4b. Provide your own thoughtful response.

    4c. 101.62

    5a. Age

    5b. NPA

    5c. 2, 12, 22, 32, 42, 52, 62, 72, 82, 92, 102

    5e. r = 0.814

    5f. At the 5% level of significance, there is a significant correlation between age and NPA (\(t\) = 4.2, \(p\) = 0.002, \(n\) = 11).

    Chapter 8

    1a. Test for Homogeneity

    1b. Goodness of Fit
    1c. Test for Independence

    1d. Goodness of Fit

    2.

    \(X = x\) 0 1 2 3
    \(P(X = x)\) 0.5787 0.34722 0.06944 0.00463

    Goodness of Fit

    \(\chi^2 = 3.43\)

    At the 5% level of significance, there is not a significant difference between the observed and expected distributions (\(\chi^2 = 3.43\), \(p > 0.1\), \(n = 158\)) . Calculator p-value is 0.33.

    3. Test for Independence
    \(\chi^2\) = 13.27
    At the 0.1 level of significance there is a correlation between shots and goals (\(\chi^2 = 13.27\), \(p < 0.005\), \(n = 49\)) (calculator p-value = \(2.696 \times 10^{-4}\)).

    4. Test for Homogeneity

    At the 0.1 level of significance, the distributions for improvement from drug and non-drug treatments are homogeneous (\(\chi^2 = 0.8222\), \(p > 0.1\), \(n = 80\)) (Calculator: \(p = 0.6629\), df=2)
    5. \(\chi^2 = 9.426\)


    6a. stratified

    6b. 2042, 584, _____

    6c. Goodness of Fit

    6d.

    屏幕快照 2019-05-23 下午1.28.36.png

    6f. At the 5% level of significance, the racial distribution of WA prisons is significantly different than what would be expected (\(\chi^2 = 229.96\), \(p < 0.005\), \(n = 300\)) (Calculator: \(p = 1.3 \times 10^{-48}\), df = 4).