Answer:
Option D is correct.
90%: (0.981, 0.1337); 95%: (0.0953, 0.1362); 99%: (0.0878, 0.1402)
Step-by-step explanation:
Confidence Interval for the population proportion is basically an interval of range of values where the true population proportion can be found with a certain level of confidence.
Mathematically,
Confidence Interval = (Sample proportion) ± (Margin of error)
Margin of Error is the width of the confidence interval about the mean.
It is given mathematically as,
Margin of Error = (Critical value) × (standard Error)
The standard error of the mean should be the same for all these given confidence intervals
But, with the sample result that should have been provided to solve this not provided, we can use mathematical methods and the right assumptions to obtain the right interval.
We first assume the sample size is large enough to make the use of z-distribution for the critical value.
90% critical value = 1.645
95% critical value = 1.960
99% critical value = 2.330
We can obtain the margin of error for each of the intervals given and since the standard error is the same for all three intervals, we can use the value of the critical value to check which intervals are the most correct.
a. 90%: (0.0962, 0.1358); 95%: (0.0919, 0.1397); 99%: (0.0899, 0.1411)
Margin of error is half the total width of the confidence interval
Margin of error of 90%: (0.0962, 0.1358) = (0.1358-0.0962)/2 = 0.0198
With critical value = 1.645
0.0198 = 1.645 × (standard error)
Standard error = 0.01204
Check if this works for the 95%: (0.0919, 0.1397)
Margin of error = 1.96 × 0.01204 = 0.02359
Margin of error calculated from the interval = (0.1397-0.0919)/2 = 0.0239
Trying the 99%: (0.0899, 0.1411)
Margin of error = 2.33 × 0.01204 = 0.02805
Margin of error calculated from the interval = (0.1411-0.0899)/2 = 0.0256
b. 90%: (0.0993, 0.1327); 95%: (0.0962, 0.1358); 99%: (0.0899, 0.1421)
Margin of error calculated from interval given = (0.1327-0.0993)/2 = 0.0167
standard error = (0.0167/1.645) = 0.010152
95%: (0.0962, 0.1358)
Margin of error = 1.96 × 0.010152 = 0.0199
Margin of error calculated from the interval = (0.1358-0.0962)/2 = 0.0198
99%: (0.0899, 0.1421)
Margin of error = 2.33 × 0.010152 = 0.02365
Margin of error calculated from the interval = (0.1421-0.0899)/2 = 0.0261
c. 90%: (0.0893, 0.1350); 95%: (0.0851, 0.1358); 99%: (0.0799, 0.1390)
90%: (0.0893, 0.1350)
Margin of Error = (0.1350-0.0893)/2 = 0.02285
Standard error = (0.02285/1.645) = 0.01389
95%: (0.0851, 0.1358);
Margin of error = 1.96 × 0.01389 = 0.02723
Margin of error calculated from the interval = (0.1358-0.0851)/2 = 0.02535
99%: (0.0799, 0.1390)
Margin of error = 2.33 × 0.01389 = 0.03236
Margin of error calculated from the interval = (0.1390-0.0799)/2 = 0.02955
d. 90%: (0.0981, 0.1337); 95%: (0.0953, 0.1362); 99%: (0.0878, 0.1402)
90%: (0.0981, 0.1337)
Margin of Error = (0.1337-0.0981)/2 = 0.0178
Standard error = (0.0178/1.645) = 0.010821
95%: (0.0953, 0.1362);
Margin of error = 1.96 × 0.010821 = 0.02045
Margin of error calculated from the interval = (0.1362-0.0953)/2 = 0.02045
99%: (0.0878, 0.1402)
Margin of error = 2.33 × 0.010821 = 0.02521
Margin of error calculated from the interval = (0.1402-0.0878)/2 = 0.0252
From this, it is evident that it is the option D that suits the mathematical answers for the co.fidemce interval.
Hope this Helps!!!
