Part One: Resting Heart Rate • Name:

The resting heart rate in beats per minute for the ESS 101j Joggling students are shown in the table.

RHR/bpm
62
63
65
68
68
68
69
69
71
72
77
79
79
82
83
86
88
91
94
94

For the resting heart rate data:

  1. _________ What level of measurement is the data?
  2. _________ Determine the sample size n.
  3. _________ Calculate the sample mean x.
  4. _________ Determine the median.
  5. _________ Determine the mode.
  6. _________ Determine the minimum.
  7. _________ Determine the maximum.
  8. _________ Calculate the range.
  9. _________ Calculate the sample standard deviation sx.
  10. _________ Calculate the sample Coefficient of Variation.
  11. _________ Determine the class width. Use 5 bins (classes or intervals)
  12. Fill in the following table with the class upper limits in the first column, the frequencies in the second column, and the relative frequencies in the third column
    BinsFrequencyRelative Frequency f/n
    ___________________________
    ___________________________
    ___________________________
    ___________________________
    ___________________________
    Sums:__________________
  13. Sketch a histogram of the relative frequency data on the back of the paper.
  14. __________________ What is the shape of the distribution?
  15. p(rhr ≤ 74.8) = _________ Using the relative frequencies in the table above, what is the probability that student will have a resting heart rate of less or equal to 74.8?
  16. Construct a 95% confidence interval for the population mean resting heart rate. Note that n is less than 30. Use the sample size, sample mean and sample standard deviation from questions two, three, and nine above to generate your t-critical tc and error tolerance E.
    1. __________ What is the point estimate for the population mean ?
    2. df = __________ How many degrees of freedom?
    3. tc = __________ What is tc?
    4. The error tolerance E = _______________
    5. The 95% confidence interval for the mean resting heart rate is ____________ ≤ ≤ ____________
  17. __________ The mean resting heart rate for human beings is commonly taken to be 72 bpm. Based on the above data and a 95% confidence level, is the mean for the Joggling class students statistically significantly different at a confidence level of 95% from 72 bpm?
  18. Perform a hypothesis test on the resting heart rate data. Test the null hypothesis that the Joggling students resting heart rate came from a population with a population mean resting heart rate of 72 bpm at a significance level of 5%.
    1. Write the null hypothesis:
    2. Write the alternate hypothesis:
    3. alpha α = __________ Write down the level of significance.
    4. tc = __________ What is tc?
    5. t = __________ Using 72 bpm as the population mean and the sample size, sample mean and sample standard deviation from questions two, three and nine, calculate the t-statistic.
    6. p = __________ Determine the p-value using the t-distribution.
    7. __________ What is the largest confidence interval c for which this difference is statistically significant?
    8. ________________________________________ Would we reject the null hypothesis or fail to reject the null hypothesis that a sample with a size specified in question two, a mean from question three, and a standard deviation as found for question nine could have come from a population with a mean of 72 bpm at a 5% level of significance?
    9. __________ If we reject the null hypothesis, what is the risk of a type I error based on the p-value?
    10. __________ If we had chosen to use an alpha α = 0.10, would the difference between the sample mean resting heart rate and the expected population mean resting heart rate of 72 have been significant?

Part Two: Regression and Correlation

Part two explores whether there is a relationship for the Joggling class students between their resting heart rates (RHR)in beats per minute and their body fat index (BFI).

RHRBFI
6210.8
6314.5
6516.4
6805.5
6811.7
6830.5
6911.3
6921.7
7123.7
7233.5
7723.5
7920.6
7931.2
8217.8
8345.5
8634.9
8838.1
9121.8
9434.0
9445.9
  1. _________ Calculate the slope of the best fit (least squares) line for the data.
  2. _________ Calculate the y-intercept of the best fit (least squares) line.
  3. _________ Is the correlation positive, negative, or neutral?
  4. _________ Use the equation of the best fit line to calculate the predicted body fat index for a resting heart rate of 74 beats per minute.
  5. _________ Use the inverse of the best fit line equation to calculate the predicted resting heart rate for a body fat index of 27.
  6. _________ Calculate the linear correlation coefficient r for the data.
  7. _________ Is the correlation none, low, moderate, high, or perfect?
  8. _________ Calculate the coefficient of determination.
  9. _________ What percent of the variation in the resting heart rate data explains the variation in the body fat index data?
  10. _________ Is there a relationship between the resting heart rate and the body fat index?
  11. The division biologist has a resting heart rate of 49 beats per minute.
    1. Can the body fat index for the division biologist be predicted?
    2. Why or why not?
    3. In terms of likely body fat index, is the division biologist likely to be (multiple choice):
      1. ___ Fit?
      2. ___ Not fit?
      3. ___ Not determinable.

Tables of Formulas and Excel functions

Basic Statistics
Statistic or Parameter Symbol Equations Excel
Square root =SQRT(number)
Sample size n =COUNT(data)
Sample mean x Σx/n =AVERAGE(data)
Sample standard deviation sx or s =STDEV(data)
Sample Coefficient of Variation CV sx/x =STDEV(data)/AVERAGE(data)
Linear Regression Statistics
Statistic or Parameter Symbol Equations Excel
Slope b =SLOPE(y data, x data)
Intercept a =INTERCEPT(y data, x data)
Correlation r =CORREL(y data, x data)
Coefficient of Determination r2 =(CORREL(y data, x data))^2
Statistic or Parameter Symbol Equations Excel
Normal Statistics
Calculate a z value from an x z = standardize.gif (905 bytes) =STANDARDIZE(x, , σ)
Calculate an x value from a z x = σ z + =σ*z+
Calculate an x from a z xbarfromz.gif (1060 bytes) = + zc*sx/sqrt(n)
Find a probability p from a z value =NORMSDIST(z)
Find a z value from a probability p =NORMSINV(p)
Confidence interval statistics
Degrees of freedom df = n-1 =COUNT(data)-1
Find a zc value from a confidence level c zc =ABS(NORMSINV((1-c)/2))
Find a tc value from a confidence level c tc =TINV(1-c,df)
Calculate an error tolerance E of a mean for n ≥ 30 using sx E error_tolerance_zc.gif (989 bytes) =zc*sx/SQRT(n)
Calculate an error tolerance E of a mean for n < 30 using sx. Should also be used for n ≥ 30. E error_tolerance_tc.gif (989 bytes) =tc*sx/SQRT(n)
Calculate a confidence interval for a population mean µ from a sample mean x and an error tolerance E x-E≤ ≤x+E
Hypothesis Testing
Calculate t-critical for a two-tailed test tc =TINV(α,df)
Calculate a t-statistic t xbartot.gif (1028 bytes) =(x - )/(sx/SQRT(n))
Calculate a two-tailed p-value from a t-statistic p = TDIST(ABS(t),df,2)