Using this sample result and a theory-based approach (and a Theory Based Inference applet), which of the following represents a three confidence intervals (90%, 95%, 99%) for the probability that a randomly selected American adult had played a Sudoku puzzle in the past year?

a. 90%: (0.0962, 0.1358); 95%: (0.0919, 0.1397); 99%: (0.0899, 0.1411)
b. 90%: (0.0993, 0.1327); 95%: (0.0962, 0.1358); 99%: (0.0899, 0.1421)
c. 90%: (0.0893, 0.1350); 95%: (0.0851, 0.1358); 99%: (0.0799, 0.1390)
d. 90%: (0.981, 0.1337); 95%: (0.0953, 0.1362); 99%: (0.0878, 0.1402)

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!!!