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    Hypothesis testing and regression analysis

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    PART I. HYPOTHESIS TESTING

    PROBLEM 1 A certain brand of fluorescent light tube was advertised as having an effective life span before burning out of 4000 hours. A random sample of 84 bulbs was burned out with a mean illumination life span of 1870 hours and with a sample standard deviation of 90 hours. Construct a 95 confidence interval based on this sample and be sure to interpret this interval.

    PROBLEM 2 Given the following data from two independent data sets, conduct a one-tail hypothesis test to determine if the means are statistically equal using alpha=0.05. Do NOT do a confidence interval.

    n1 = 35 n2 = 30
    xbar1= 32 xbar2 = 25
    s1=7 s2 = 6

    PROBLEM 3. A test was conducted to determine whether gender of a display model affected the likelihood that consumers would prefer a new product. A survey of consumers at a trade show which used a female spokesperson determined that 120 of 300 customers preferred the product while 92 of 280 customers preferred the product when it was shown by a female spokesperson.

    Do the samples provide sufficient evidence to indicate that the gender of the salesperson affect the likelihood of the product being favorably regarded by consumers? Evaluate with a two-tail, alpha =.01 test. Do NOT do a confidence interval.

    PROBLEM 4 Asuming that the population variances are equal for Male and Female GPA's, test the following sample data to see if Male and Female PhD candidate GPA's (Means) are equal. Conduct a two-tail hypothesis test at α =.01 to determine whether the sample means are different. Do NOT do a confidence interval.

    Male GPA's Female GPA's
    Sample Size 12 13
    Sample Mean 2.8 4.95
    Sample Standard Dev .25 .8

    PART II REGRESSION ANALYSIS
    Problem 5 You wish to run the regression model (less Intercept and coefficients) shown below:
    VOTE = URBAN + INCOME + EDUCATE

    Given the Excel spreadsheet below for annual data from1970 to 2006 (with the data for row 5 thru row 35 not shown), complete all necessary entries in the Excel Regression Window shown below the data.

    A B C D E
    1 YEAR VOTE URBAN INCOME EDUCATE
    2 1970 49.0 62.0 7488 4.3
    3 1971 58.3 65.2 7635 8.3
    4 1972 45.2 75.0 7879 4.5

    36 2004 50.1 92.1 15321 4.9
    37 2005 67.7 94.0 15643 4.7
    38 2006 54.2 95.6 16001 5.1

    Regression

    Input OK
    Input Y Range:
    Cancel
    Input X Range:
    Help
    Labels Constant is Zero
    Confidence Level: 95 %

    Output options
    Output Range:

    New Worksheet Ply:
    New Workbook

    Residuals
    Residuals Residual Plots
    Standardized Residuals Line Fit Plots

    Normal Probability
    Normal Probability Plots

    PROBLEM 6. Use the following regression output to determine the following:

    A real estate investor has devised a model to estimate home prices in a new suburban development. Data for a random sample of 100 homes were gathered on the selling price of the home ($ thousands), the home size (square feet), the lot size (thousands of square feet), and the number of bedrooms.

    The following multiple regression output was generated:

    Regression Statistics
    Multiple R 0.8647
    R Square 0.7222
    Adjusted R Square 0.6888
    Standard Error 16.0389
    Observations 100

    Coefficients Standard Error t Stat P-value
    Intercept -24.888 38.3735 -0.7021 0.2154
    X1 (Square Feet) 0.2323 0.0184 9.3122 0.0000
    X2 (Lot Size) 11.2589 1.7120 4.3256 0.0001
    X3 (Bedrooms) 15.2356 6.8905 3.2158 0.1589

    a. Why is the coefficient for BEDROOMS a positive number?

    b. Which is the most statistically significant variable? What evidence shows this?

    c. Which is the least statistically significant variable? What evidence shows this?

    d. For a 0.05 level of significance, should any variable be dropped from this model? Why or why not?

    e. Interpret the value of R squared? How does this value from the adjusted R squared?

    f. Predict the sales price of a 1134-square-foot home with a lot size of 15,400 square feet and 2 bedrooms.

    PART III SPECIFIC KNOWLEDGE SHORT-ANSWER QUESTIONS.

    Problem 7 Define Autocorrelation in the following terms:
    a. In what type of regression is it likely to occur?

    b. What is bad about autocorrelation in a regression?

    c. What method is used to determine if it exists? (Think of statistical test to be used)

    d. If found in a regression how is it eliminated?

    Problem 8 Define Multicollinearity in the following terms:
    a. In what type of regression is it likely to occur?

    b. Why is multicollinearity in a regression a difficulty to be resolved?

    c. How can multicollinearity be determined in a regression?.

    d. If multicollinearity is found in a regression, how is it eliminated?

    STUDENT T TABLE

    df .10 .05 .025 .010 .005

    1 3.078 6.314 12.706 31.821 63.657
    2 1.886 2.920 4.303 6.965 9.925
    3 1.638 2.353 3.182 4.541 5.841
    4 1.533 2.132 2.776 3.747 4.604
    5 1.476 2.015 2.571 3.365 4.032
    6 1.440 1.943 2.447 3.143 3.707
    7 1.415 1.895 2.365 2.998 3.499
    8 1.397 1.860 2.306 2.896 3.355
    9 1.383 1.833 2.262 2.821 3.250
    10 1.372 1.812 2.228 2.764 3.169
    11 1.363 1.796 2.201 2.718 3.106
    12 1.356 1.782 2.179 2.681 3.055
    13 1.350 1.771 2.160 2.650 3.012
    14 1.345 1.761 2.145 2.624 2.977
    15 1.341 1.753 2.131 2.602 2.947
    16 1.337 1.746 2.120 2.583 2.921
    17 1.333 1.740 2.110 2.567 2.898
    18 1.330 1.734 2.101 2.552 2.878
    19 1.328 1.729 2.093 2.539 2.861
    20 1.325 1.725 2.086 2.528 2.845
    21 1.323 1.721 2.080 2.518 2.831
    22 1.321 1.717 2.074 2.508 2.819
    23 1.319 1.714 2.069 2.500 2.807
    24 1.318 1.711 2.064 2.492 2.797
    25 1.316 1.708 2.060 2.485 2.787
    26 1.315 1.706 2.056 2.479 2.779
    27 1.314 1.703 2.052 2.473 2.771
    28 1.313 1.701 2.048 2.467 2.763
    29 1.311 1.699 2.045 2.462 2.756
    30 1.310 1.697 2.042 2.457 2.750
    40 1.303 1.684 2.021 2.423 2.704
    60 1.296 1.671 2.000 2.390 2.660
    120 1.289 1.658 1.980 2.358 2.617
     1.282 1.645 1.960 2.326 2.576

    DURBIN-WATSON d STATISTIC, = .05

    n k=1 k=2 k=3 k=4 k=5
    dL dU dL dU dL dU dL dU dL dU
    15 1.08 1.36 .95 1.54 .82 1.75 .69 1.97 .56 2.21
    16 1.10 1.37 .98 1.54 .86 1.73 .74 1.93 .62 2.15
    17 1.13 1.38 1.02 1.54 .90 1.71 .78 1.90 .67 2.10
    18 1.16 1.39 1.05 1.53 .93 1.69 .82 1.87 .71 2.06
    19 1.18 1.40 1.08 1.53 .97 1.68 .86 1.85 .75 2.02
    20 1.20 1.41 1.10 1.54 1.00 1.68 .90 1.83 .79 1.99
    21 1.22 1.42 1.13 1.54 1.03 1.67 .93 1.81 .83 1.96
    22 1.24 1.43 1.15 1.54 1.05 1.66 .96 1.80 .86 1.94
    23 1.26 1.44 1.17 1.54 1.08 1.66 .99 1.79 .90 1.92
    24 1.27 1.45 1.19 1.55 1.10 1.66 1.01 1.78 .93 1.90
    25 1.29 1.45 1.21 1.55 1.12 1.66 1.04 1.77 .95 1.89
    26 1.30 1.46 1.22 1.55 1.14 1.65 1.06 1.76 .98 1.88
    27 1.32 1.47 1.24 1.56 1.16 1.65 1.08 1.76 1.01 1.86
    28 1.33 1.48 1.26 1.56 1.18 1.65 1.10 1.75 1.03 1.85
    29 1.34 1.48 1.27 1.56 1.20 1.65 1.12 1.74 1.05 1.84
    30 1.35 1.49 1.28 1.57 1.21 1.65 1.14 1.74 1.07 1.83
    31 1.36 1.50 1.30 1.57 1.23 1.65 1.16 1.74 1.09 1.83
    32 1.37 1.50 1.31 1.57 1.24 1.65 1.18 1.73 1.11 1.82
    33 1.38 1.51 1.32 1.58 1.26 1.65 1.19 1.73 1.13 1.81
    34 1.39 1.51 1.33 1.58 1.27 1.65 1.21 1.73 1.15 1.81
    35 1.40 1.52 1.34 1.58 1.28 1.65 1.22 1.73 1.16 1.80
    36 1.41 1.52 1.35 1.59 1.29 1.65 1.24 1.73 1.18 1.80
    37 1.42 1.53 1.36 1.59 1.31 1.66 1.25 1.72 1.19 1.80
    38 1.43 1.54 1.37 1.59 1.32 1.66 1.26 1.72 1.21 1.79
    39 1.43 1.54 1.38 1.60 1.33 1.66 1.27 1.72 1.22 1.79
    40 1.44 1.54 1.39 1.60 1.34 1.66 1.29 1.72 1.23 1.79
    45 1.48 1.57 1.43 1.62 1.38 1.67 1.34 1.72 1.29 1.78
    50 1.50 1.59 1.46 1.63 1.42 1.67 1.38 1.72 1.34 1.77
    55 1.53 1.60 1.49 1.64 1.45 1.68 1.41 1.72 1.38 1.77
    60 1.55 1.62 1.51 1.65 1.48 1.69 1.44 1.73 1.41 1.77
    65 1.57 1.63 1.54 1.66 1.50 1.70 1.47 1.73 1.44 1.77
    70 1.58 1.64 1.55 1.67 1.52 1.70 1.49 1.74 1.46 1.77
    75 1.60 1.65 1.57 1.68 1.54 1.71 1.51 1.74 1.49 1.77
    80 1.61 1.66 1.59 1.69 1.56 1.72 1.53 1.74 1.51 1.77
    85 1.62 1.67 1.60 1.70 1.57 1.72 1.55 1.75 1.52 1.77
    90 1.63 1.68 1.61 1.70 1.59 1.73 1.57 1.75 1.54 1.78
    95 1.64 1.69 1.62 1.71 1.60 1.73 1.58 1.75 1.56 1.78
    100 1.65 1.69 1.63 1.72 1.61 1.74 1.59 1.76 1.57 1.78

    MEAN HYPOTHESIS TEST AND CONFIDENCE INTERVAL FORMULAS

    Null Hypothesis Standard Deviation Data "t" for Hypothesis Test Confidence Interval

    ["Previous Standard" is "old"
    numerical constant to which
    new statistic is being compared.]

    DF = n-1

    Standard Deviation
    is the "data" t
    Denominator

    LARGE SAMPLE DF =

    Use "pooled s" above in small sample t and confidence interval formulas.
    SMALL SAMPLE: DF = n1 + n2 - 2
    ******************************************************************************************************************************************
    Determine Sample size for when B = Error of Estimation

    ALWAYS ROUND UP VALUE OF n DETERMINED IN FORMULA

    LARGE SAMPLE: DF =

    BINOMIAL PROBABILITY HYP TEST & CONFIDENCE INTERVAL FORMULAS

    Null Hypothesis Estimator "Data" t for Hypothesis Test Confidence Interval

    Ho: p = Previous Probability Standard

    [Previous Probability Standard
    is "old" probability to which
    probability based on new data
    is being compared.]

    ************************************************************************************************************
    H0 :
    (Previous Probability
    Difference assumed
    to be 0.)

    ************************************************************************************************************
    DETERMINE SAMPLE SIZE FOR BINOMIAL p
    B = Error of Estimation
    [Round sample size "n" determined in this formula to next higher integer.]

    DF for sample size is always is value closest to .5

    © BrainMass Inc. brainmass.com October 9, 2019, 11:05 pm ad1c9bdddf
    https://brainmass.com/math/interpolation-extrapolation-and-regression/hypothesis-testing-and-regression-analysis-244029

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    Solution Summary

    The solution provides step by step method for the calculation of regression model and test statistic for hypothesis testing problems . Formula for the calculation and Interpretations of the results are also included. Interactive excel sheet is included. The user can edit the inputs and obtain the complete results for a new set of data.

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