Validate Survey Accuracy with Confidence Intervals

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Business Context

InsightLoop, a customer analytics company, tracks whether its survey-based satisfaction estimates are accurate enough to support quarterly product decisions. An analyst wants to verify that the latest estimate is statistically reliable before presenting it to leadership.

Problem Statement

A random sample of customers was surveyed, and the analyst observed the share who reported being satisfied. Your task is to assess the accuracy of this analysis using a confidence interval and a hypothesis test against the companys historical benchmark.

Given Data

Metric	Value
Total customers surveyed	400
Customers reporting satisfaction	232
Sample satisfaction rate	58.0%
Historical satisfaction benchmark	55.0%
Significance level	5.0%

Requirements

Compute the sample proportion of satisfied customers.
Construct a 95% confidence interval for the true satisfaction rate.
Test whether the current satisfaction rate is different from the historical benchmark of 55% using a two-sided one-sample proportion z-test.
Report the z-statistic and p-value.
State whether the result is statistically significant at $\alpha = 0.05$ .
Explain what the interval and test say about the accuracy and decision usefulness of the analysis.

Assumptions

The 400 customers were randomly sampled.
Each survey response is independent.
The normal approximation is appropriate because $np$ and $n(1-p)$ are both comfortably above 10.
Nonresponse bias and survey wording effects are not modeled here.