Estimating a Proportion: Confidence, Precision and a Population Claim — Solutions and Rubrics

p is the proportion of all 6,000 enrolled students who prefer the proposed menu. Use a one-sample z-interval for p. SRS gives randomization. Without replacement, 200≤0.10(6,000)=600. Observed successes n p̂=120 and failures n(1−p̂)=80 are each at least 10. Both count checks use observed p̂=0.60 because the interval estimates unknown p rather than assuming a null value.

p̂=120/200=0.60. SE=√[0.60(0.40)/200]=0.0346410. MOE=1.96(0.0346410)=0.0678964. Interval p̂±MOE=(0.5321036,0.6678964). Standard normal boundaries −1.96 and +1.96 enclose approximately the central 95% area. SE estimates the sampling-distribution SD of p̂ and typical sample-proportion variation from population p; MOE is the critical multiplier times SE. Equal extension in each direction makes MOE half the full width 0.1357928.

We are 95% confident that the population proportion of enrolled students preferring the proposed menu is between about 0.532 and 0.668. In repeated SRSs of 200 using the same method, about 95% of constructed intervals contain the fixed population proportion. A particular interval may or may not contain p. Since every value in this interval exceeds 0.50, it provides convincing evidence for a population majority preference. The interval estimates one population proportion; individual responses are prefer/not prefer, not proportions to place between endpoints.

From MOE=z*√[p̂(1−p̂)/n], solve n=(z*)²p̂(1−p̂)/(MOE)². Pilot planning: n=1.96²(0.60)(0.40)/0.04²=576.24, so use at least 577. With no estimate, p̂=0.50 maximizes p̂(1−p̂)=0.25; n=600.25, so use 601. Both are within 10% of 200,000. Rounding up ensures the margin is no greater than the target; rounding down can make it slightly too large. These are planning estimates, not a substitute for later design/count checks.

SE=√(0.24/400)=0.0244949. MOEs are approximately 0.0402941 at 90%, 0.0480100 at 95% and 0.0630989 at 99%; widths are twice these, approximately 0.0805882, 0.0960200 and 0.1261977. Higher confidence captures more of the theoretical sampling distribution, requiring a larger critical value and therefore larger MOE and width for the same sample.

SE400≈0.0244949 and width≈0.0960200. SE1600=√(0.24/1,600)≈0.0122474 and width=2(1.96)(0.0122474)≈0.0480100. Quadrupling n halves both SE and width when the confidence multiplier and estimated proportion stay fixed. Width is proportional to 1/√n under these settings, not 1/n; larger n reduces typical sampling variation.

We are 90% confident that the population support proportion lies from about 0.4080 to 0.5920; in this simulation we can check that 0.55 is inside. A particular computed interval either contains the fixed parameter or does not, so some intervals miss. The 90% level describes the approximate long-run fraction of intervals capturing p in repeated same-size sampling, not a guarantee that exactly 90 of every 100 capture. Observed 88/100 is one finite-run capture proportion, not a redefinition of the preset procedure.

The interval contains proportions below, equal to and above 0.50. A majority is plausible but not established by an interval entirely above 0.50, so this interval does not provide convincing evidence for majority support. It also contains 0.55 and other majority values, so it does not prove the absence of a majority. Plausibility and a directional evidentiary claim are different.

The parameter is the solar-owner proportion in each stated target household population. A has randomization and 60≤400, but observed counts 4 and 56 fail the at-least-10 success requirement. B has observed counts 500/500 but volunteers do not justify randomization; unknown population/design leave the sampling-fraction condition unsupported. C has randomization and counts 70/50, but 120>0.10(900)=90 fails the without-replacement check. An interval estimates an unknown p, so its count condition uses observed success/failure counts, not a convenient inserted null p₀.