95% prediction interval for price $10,167 $16,771 95% confidence interval for mean price $11,793 $15,145
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!
In Exercises, presume that the assumptions for regression inferences are met.
Tax Efficiency. Following are the data on percentage of investments in energy securities and tax efficiency from Exercise.
| x | 3.1 | 3.2 | 3.7 | 4.3 | 4.0 | 5.5 | 6.7 | 7.4 | 7.4 | 10.6 |
| y | 98.1 | 94.7 | 92.0 | 89.8 | 87.5 | 85.0 | 82.0 | 77.8 | 72.1 | 53.5 |
a. Obtain a point estimate for the mean tax efficiency of all mutual fund portfolios with 6% of their investments in energy securities.
b. Determine a 95% confidence interval for the mean tax efficiency of all mutual fund portfolios with 6% of their investments in energy securities.
c. Find the predicted tax efficiency of a mutual fund portfolio with 6% of its investments in energy securities.
d. Determine a 95% prediction interval for the tax efficiency of a mutual fund portfolio with 6% of its investments in energy securities.
e. Draw graphs similar to those in Fig. on page 683, showing both the 95% confidence interval from part (b) and the 95% prediction interval from part (d).
f. Why is the prediction interval wider than the confidence interval?
FIGURE Prediction and confidence intervals for 3-year-old Orions
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