A researcher went out and collected a random sample of 25 individual incomes. The statistics he calculated were: The sample mean income is $39,600 a year The sample standard deviation is $4,240 a year Build a 95% confidence interval for the population mean income. Show your work, explain your choices. Test H0 that the population mean income is $38,000 a year, at a 5% significance level. Show your work, explain your choices. Given your work in part (B) above, would you be correct to say, “There is sufficient evidence to conclude that the population mean income is greater than $38,000 a year?” Explain why or why not.
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!
Some critical values you may want to use:
- tα, d.f. is the value such that probability P(t > tα, d.f.) = α, where t is a variable that follows a Student’s t-distribution with d.f. degrees of freedom: t0.05, 24 = 1.711; t0.05, 25 = 1.708; t0.025, 24 = 2.064; t0.025, 25 = 2.06; t0.1, 24 = 1.318; t0.1, 25 = 1.316.
- Zα is the value such that probability P(Z > Zα) = α, where Z is a variable that follows a standard
normal distribution: Z0.05 = 1.645; Z0.025 = 1.96; Z0.1 = 1.282.
A researcher went out and collected a random sample of 25 individual incomes. The statistics he calculated were:
- The sample mean income is $39,600 a year
- The sample standard deviation is $4,240 a year
- Build a 95% confidence interval for the population mean income. Show your work, explain your choices.
- Test H0 that the population mean income is $38,000 a year, at a 5% significance level. Show your work, explain your choices.
- Given your work in part (B) above, would you be correct to say, “There is sufficient evidence to conclude that the population mean income is greater than $38,000 a year?” Explain why or why not.
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