et x be a random variable that represents the percentage of successful free throws a professional basketball player makes in a season. Let y be a random variable that represents the percentage of successful field goals a professional basketball player makes in a season. A random sample of n = 6 professional basketball players gave the following information. x 67 64 75 86 73 73 y 42 39 48 51 44 51 (a) Verify that Σx = 438, Σy = 275, Σx2 = 32264, Σy2 = 12727, Σxy = 20231, and r ≈ 0.827. Σx Σy Σx2 Σy2 Σxy r (b) Use a 5% level of significance to test the claim that ? > 0. (Round your answers to two decimal places.) t critical t Conclusion? Reject the null hypothesis, there is sufficient evidence that ? > 0. Reject the null hypothesis, there is insufficient evidence that ? > 0. Fail to reject the null hypothesis, there is insufficient evidence that ? > 0. Fail to reject the null hypothesis, there is sufficient evidence that ? > 0. (c) Verify that Se ≈ 3.1191, a ≈ 6.564, b ≈ 0.5379, and x ≈ 73.000. Se a b x (d) Find the predicted percentage of successful field goals for a player with x = 73% successful free throws. (Round your answer to two decimal places.) % (e) Find a 90% confidence interval for y when x = 73. (Round your answers to one decimal place.) lower limit % upper limit % (f) Use a 5% level of significance to test the claim that ? > 0. (Round your answers to two decimal places.) t critical t Conclusion Reject the null hypothesis, there is sufficient evidence that ? > 0. Reject the null hypothesis, there is insufficient evidence that ? > 0. Fail to reject the null hypothesis, there is insufficient evidence that ? > 0. Fail to reject the null hypothesis, there is sufficient evidence that ? > 0 (g) Find a 90% confidence interval for ?. (Round your answers to three decimal places.) lower limit upper limit Interpret its meaning. For every percentage increase in successful free throws, the percentage of successful field goals increases by an amount that falls within the confidence interval.
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!
Let x be a random variable that represents the percentage of successful free throws a professional basketball player makes in a season. Let y be a random variable that represents the percentage of successful field goals a professional basketball player makes in a season. A random sample of n = 6 professional basketball players gave the following information.
| x | 67 | 64 | 75 | 86 | 73 | 73 |
| y | 42 | 39 | 48 | 51 | 44 | 51 |
(a) Verify that Σx = 438, Σy = 275, Σx2 = 32264, Σy2 = 12727, Σxy = 20231, and r ≈ 0.827.
| Σx | |
| Σy | |
| Σx2 | |
| Σy2 | |
| Σxy | |
| r |
(b) Use a 5% level of significance to test the claim that ? > 0. (Round your answers to two decimal places.)
| t | |
| critical t |
Conclusion?
(c) Verify that Se ≈ 3.1191, a ≈ 6.564, b ≈ 0.5379, and x ≈ 73.000.
| Se | |
| a | |
| b | |
| x |
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(d) Find the predicted percentage of successful field goals for a player with x = 73% successful free throws. (Round your answer to two decimal places.)
%
(e) Find a 90% confidence interval for y when x = 73. (Round your answers to one decimal place.)
| lower limit | % |
| upper limit | % |
(f) Use a 5% level of significance to test the claim that ? > 0. (Round your answers to two decimal places.)
| t | |
| critical t |
Conclusion
| lower limit | |
| upper limit |
Interpret its meaning.
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