In: Statistics and Probability
For this problem, carry at least four digits after the decimal
in your calculations. Answers may vary slightly due to
rounding.
Santa Fe black-on-white is a type of pottery commonly found at
archaeological excavations at a certain monument. At one excavation
site a sample of 594 potsherds was found, of which 359 were
identified as Santa Fe black-on-white.
(a) Let p represent the proportion of Santa Fe
black-on-white potsherds at the excavation site. Find a point
estimate for p. (Round your answer to four decimal
places.)
(b) Find a 95% confidence interval for p. (Round your
answers to three decimal places.)
lower limit | |
upper limit |
Give a brief statement of the meaning of the confidence
interval.
5% of the confidence intervals created using this method would include the true proportion of potsherds.5% of all confidence intervals would include the true proportion of potsherds. 95% of the confidence intervals created using this method would include the true proportion of potsherds.95% of all confidence intervals would include the true proportion of potsherds.
(c) Do you think that np > 5 and nq > 5 are
satisfied for this problem? Explain why this would be an important
consideration.
Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately normal.Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately binomial. No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately normal.No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately binomial.
Solution :
Given that,
n = 594
x = 359
a) Point estimate = sample proportion = = x / n = 359 / 594 = 0.6044
1 - = 1 - 0.6044 = 0.3956
b) At 95% confidence level the z is,
= 1 - 95%
= 1 - 0.95 = 0.05
/ 2 = 0.025
Z/2 = Z 0.025 = 1.96
Margin of error = E = Z / 2 * (( * (1 - )) / n)
= 1.96 * (((0.6044* 0.3956) / 594 )
= 0.039
A 95% confidence interval for population proportion p is ,
- E < p < + E
0.6044 - 0.039 < p < 0.6044 + 0.039
0.565 < p < 0.643
The 95% confidence interval for the population proportion p is : Lower limit = 0.565, upper limit = 0.643
95%of the confidence interval created using this method would include the true proportion of potsherds.
c) Yes, the conditions are satisfied. This is important because it allows us to say that is approximately normal.