The heights, in inches, for 120 people (male and femae) have been measured. The
ID: 3072358 • Letter: T
Question
The heights, in inches, for 120 people (male and femae) have been measured. The results are presented Heights sample data 64.46 62.71 67.65 69.58 67.12 69.56 68.72 66.59 68.0568.4766.47 65.42 65.22 70.59 63.24 66.09 62.34 71.05 69.37 71.76 71.37 67.07 67.88 64.61 69.8 64.4 70.67 64.24 65.39 65.0867.8460.95 66.91 71.12 61.72 66.2 66.54 64.13 69.69 68.93 76.1 63.26 66.59 63.8 63.74 67.53 69.07 65.13 69.35 64.83 66.94 67.23 66.9 65.4 67.79 70.12 60.4 66.16 69.85 64.94 62.23 64.56 63.01 69.27 69.01 64.14 69.14 69.76 66.52 70.05 68.1 64.5 65.04 62.4265.72 72.36 63.31 66.71 69.85 64.52 69.27 68.5 70.54 65.12 68.16 68.44 66.48 64.03 65.01 66.85 66.33 65.75 64.7464.7567.88 64.24 68.76 61.1162.74 69.77 64.45 71.4 71.13 70.2 67.46 66.55 71.75 63.96 65.38 70.35 65.13 67.5570.27 68.76 70.35 69 62.36 64.73 71.38 66.77 a) Calculate the mean and standard deviation for the sample. Give your answers to 2 decimal places sample mean66.93 sample standard deviation-2.89 b) Find the proportion of heights that are within 1 standard deviation of the sample mean and also the proportion that are within 2 standard deviations of the sample mean. Use the unrounded values for the mean and standard deviation when doing this calculation. Give your answers as decimals to 2 decimal places. Proportion of heights within 1 standard deviation of the mean- Proportion of heights within 2 standard deviations of the mean c) Select the appropriate description for the data the data are APPROXIMATELY normal the data are CLEARLY not normal d) Calculate the standardized value for the value 64. Note that, for a value x within a sample that is approximately distributed as N(x,s), a standardized value can be calculated as z-(x- x)/s standardized value (to 2 decimal places) for the value 64 - e) Calculate the probability that a standard normal random variable Z takes a values less than the standardized value calculated in part d). Give your answer as a decimal to 4 decimal places Probability Z less than standardized value- f) Find the proportion of values in the sample that are less than or equal to 64. Give you answer as a decimal to 2 decimal places. Proportion of values less than or equal to 64Explanation / Answer
(a). Sample mean = 66.93 ; sample standard deviation = 2.89
(b) No of sample values lying within 1 standard deviation of the mean is 81 and hence the proportion is 81 / 120 = 0.68
No of sample values lying within 1 standard deviation of the mean is 116 and hence the proportion is 116 / 120 = 0.97
(c) For a Normal variate the probaility of lying within 1 standard deviation of the mean is 0.68 and that of 2 standard deviation of the mean is 0.95 . Hence comparing with the corresponding sample proportions we can see that the data are APPROXIMATELY normal.
(d) The standardized value of 64 is (64-66.93) / 2.89 = -1.01
(e) The probability that a standard normal variable is less -1.01 is 0.15
(f) The no of sample observations lying below 64 is 17 and hence the proportion os sample points lying below 64 is 17 / 120 = 0.14
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