1. A prison official wants to estimate the proportion of cases
of recidivism. Examining the records of 238
convicts, the official determines that there are
92 cases of recidivism. Find the lower limit of
the 90% confidence interval estimate for the
population proportion of cases of recidivism.
(Round to 3 decimal places.)
2. A prison official wants to estimate the proportion of cases
of recidivism. Examining the records of 157
convicts, the official determines that there are
47 cases of recidivism.
Find the margin of error for a 90% confidence
interval estimate for the population proportion of cases of
recidivism. (Use 3 decimal places.)
3. A prison official wants to estimate the proportion of cases
of recidivism. Examining the records of 221
convicts, the official determines that there are
47 cases of recidivism. A confidence interval will
be obtained for the proportion of cases of recidivism. Part of this
calculation includes the estimated standard error of the sample
proportion.
Calculate the estimated standard error. (Use 3
decimal places in calculations and in reporting your answer.)
4. The head of a computer science department is interested in
estimating the proportion of students entering the department who
will choose the new computer engineering option. A preliminary
sample indicates that the proportion will be around
0.231.
What size sample should the department head take if she wants to be
95% confident that the estimate is within 0.02 of
the true proportion?
5. A quality control engineer is interested in estimating the
proportion of defective items coming off a production line. In a
sample of 351 items, 48 are
defective. Calculate a 95.0% confidence
interval estimate for the proportion of defectives from this
production line. (Use 3 decimal places in calculations and in
reporting your answers.)
Lower Limit: ?
Upper Limit: ?
Answer:
1.
Given,
p^ = x/n = 92/238
= 0.387
Here ata 90% CI, z value = 1.645
CI = p^ +/- z*sqrt(p^(1-p^)/n)
substitute values
= 0.387 +/- 1.645*sqrt(0.387(1-0.387)/238)
= 0.387 +/- 0.0519
= (0.3351 , 0.4389)
lower limit = 0.3351
2.
p^ = 47/157
= 0.2994
At 90% CI, z value = 1.645
Margin of error E = z*sqrt(p^(1-p^)/n)
substitute values
= 1.645*sqrt(0.2994(1-0.2994)/157)
E = 0.060
3.
Standard error = sqrt(p^(1-p^)/n)
p^ = x/n = 47/221 = 0.27
SE = sqrt(0.27(1-0.27)/221)
= 0.0299
4.
Given,
p^ = 0.231
E = 0.02
Here for 95% CI, z value is 1.96
n = p^(1-p^)*(z/E)^2
substitute values
= 0.231(1-0.231)*(1.96/0.02)^2
= 1706.044956
= 1706
5.
p^ = x/n = 48/351 = 0.1368
Here for 95% CI, z value is 1.96
CI = p^ +/- z*sqrt(p^(1-p^)/n)
substitute values
= 0.1358 +/- 1.96*sqrt(0.1358(1-0.1358)/351)
= 0.1358 +/- 0.0358
= (0.1 , 0.1716)
Lower limit = 0.1
Upper limit = 0.1716
1. A prison official wants to estimate the proportion of cases of recidivism. Examining the records...
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