Many body measurements of people of the same sex and similar ages such as height and upper arm length follow a Normal distribution reasonably closely. Weights, on the other hand, are not Normally distributed. Suppose a survey includes the weights of a representative sample of 548 females in the United States aged 20 –29 . Suppose the mean of the weights was 161.59 pounds, and the standard deviation was 48.95 pounds. The histogram of the data includes a smooth curve representing an N(161.59,48.95) distribution. From the histogram, the Normal curve does not appear to follow the pattern in the histogram that closely. Because of this, the use of areas under the Normal curve may not provide a good approximation to weights in various intervals. You may find Table A useful. (a) There were12 females that weighed under 100 pounds. What percent of females aged 20 to 29 weighed under 100 pounds? (Enter your answer rounded to two decimal places.) percent under 100 lbs: What percent of the N(161.59,48.95) distribution is below 100 ? (Enter your answer rounded to two decimal places.) percent below 100 : (b) There were 34 females that weighed over 250 pounds. What percent of females aged 20 to 29 weighed over 250 pounds? (Enter your answer rounded to two decimal places.) percent over 250 lbs: What percent of the N(161.59,48.95) distribution is above 250 ? (Enter your answer rounded to two decimal places.) percent above 250 : (c) Based on your answers in parts (a) and (b), do you think it is a good idea to summarize the distribution of weights by an N(161.59,48.95) distribution? Yes, because the predicted percentages are not significantly different from the actual percentages. No, because the predicted percentages are significantly different than the actual percentages. No, because the sample size is too small. Yes, because most females weigh between100 pounds and 250 pounds.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Many body measurements of people of the same sex and similar ages such as height and upper arm length follow a Normal distribution reasonably closely. Weights, on the other hand, are not
The histogram of the data includes a smooth curve representing an N(161.59,48.95) distribution.
From the histogram, the Normal curve does not appear to follow the pattern in the histogram that closely. Because of this, the use of areas under the Normal curve may not provide a good approximation to weights in various intervals.
You may find Table A useful.
(a) There were12 females that weighed under 100 pounds. What percent of females aged 20 to 29 weighed under 100 pounds? (Enter your answer rounded to two decimal places.)
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