Question

A study was conducted to aid in the remodeling of an office building that contains three entrances. The choice of entrance was recorded for a sample of 200 persons who entered the building. Do the data in the table indicate that there is a difference in preference for the three entrances? Find a \(95 \%\) confidence interval for the proportion of persons favoring entrance 1. $$ \begin{array}{l|ccc} \text { Entrance } & 1 & 2 & 3 \\ \hline \text { Number Entering } & 83 & 61 & 56 \end{array} $$

Short answer

Also, what is the estimated 95% confidence interval for the proportion of persons favoring entrance 1? Answer: Although the proportions of people choosing each entrance seem to differ slightly, with entrance 1 having the highest proportion (0.415), we cannot conclusively say that there is a difference in preferences without conducting a proper statistical test. The 95% confidence interval for the proportion of persons favoring entrance 1 is (0.3212, 0.5088).

Step-by-step solution

Calculate proportions for each entrance

First, calculate the proportions of people choosing each entrance by dividing the number of people entering each entrance by the total number of people (200). \( p_1 = \frac{83}{200} \) \( p_2 = \frac{61}{200} \) \( p_3 = \frac{56}{200} \)

Analyze proportions

Now, look at the proportions to check if there seems to be a difference in preferences for each entrance: \(p_1 = 0.415\) \(p_2 = 0.305\) \(p_3 = 0.280\) The proportions seem to differ slightly, with entrance 1 having the highest proportion. However, we cannot conclude there is a preference difference without a proper statistical test or confidence interval estimation.

Calculate 95% confidence interval for entrance 1

We will use the formula for confidence intervals of proportions: \( CI = p_1 \pm z \times \sqrt{\frac{p_1(1-p_1)}{n}} \) Where: - \(CI\) is the confidence interval, - \(p_1\) is the proportion of people choosing entrance 1, - \(z\) is the z-score corresponding to the desired confidence level (95%), - \(n\) is the sample size (200). For a 95% confidence interval, the z-score is approximately 1.96. Plug in the values and calculate the confidence interval: \(CI = 0.415 \pm 1.96 \times \sqrt{\frac{0.415(1-0.415)}{200}}\) \(CI = 0.415 \pm 1.96 \times \sqrt{\frac{0.415 \times 0.585}{200}}\) \(CI = 0.415 \pm 1.96 \times 0.0479\) \(CI = 0.415 \pm 0.0938\) So the 95% confidence interval for the proportion of persons favoring entrance 1 is \((0.3212, 0.5088)\). In conclusion, based on the proportions, there seems to be a slight difference in preferences for the entrances, but a proper statistical test would be required to make a definitive conclusion. The 95% confidence interval for the proportion of persons favoring entrance 1 is \((0.3212, 0.5088)\).

Key concepts

Proportion Estimation

Proportion estimation is pivotal in statistics when we want to understand the likely percentage of a population that exhibits a certain feature or behavior. In our study, the population is the group of all people who enter the office building, and the feature of interest is the preference for entrance 1.

To estimate the proportion, we divide the number of people who exhibit the feature (those who chose entrance 1) by the total number of people observed (200 in this case). This calculation provides us with a proportion, or in layman terms, an educated guess, of the overall preference based on our sample. It is important to note that the precision of our estimate depends largely on the size of the sample—the larger the sample, the more confident we can be in our estimation.

This proportion, while informative, doesn't provide a full picture of the population's behavior. To enhance our understanding, we use a confidence interval which tells us the range within which the true population proportion likely falls. This is not just one estimate, but rather a range of plausible values for the proportion that reflects the uncertainty inherent in any statistical estimation process.

Statistical Hypothesis Testing

Statistical hypothesis testing is a method used to decide whether the data at hand sufficiently supports a specific hypothesis. In the context of the office building entrances, suppose we begin with the hypothesis that there is no preference among the three entrances—the null hypothesis. We would compare this against the alternative hypothesis that there is indeed a preference.

Testing this involves assessing whether the observed differences in the proportions from our sample could be due to random chance alone, or if they are statistically significant. A common way to do this is to calculate a test statistic, which allows us to infer whether our observed sample is consistent with the null hypothesis.

If the test statistic falls within a certain critical range, it suggests that the observed data is unlikely under the null hypothesis, leading us to reject the null hypothesis in favor of the alternative. This doesn't prove that the alternative hypothesis is true, but rather that the evidence we have is strong enough to act as if it is.

Z-Score

The z-score is a measurement of how many standard deviations a data point is from the mean. In the case of our confidence interval for proportion estimation, the z-score relates to the standard normal distribution and is used to determine how 'extreme' a proportion is relative to what we would expect given the null hypothesis is true.

For a 95% confidence interval, the z-score is roughly 1.96, meaning that if the null hypothesis were true, we would expect our proportion to lie within 1.96 standard deviations from the mean 95% of the time. The calculation of confidence intervals involves this z-score to account for the level of certainty (or confidence) we wish to have in our interval estimate. In simple terms, a higher z-score would create a wider confidence interval, reflecting a higher certainty that the true population proportion lies within that range.

By using the z-score in our calculations, we more clearly communicate the idea that our estimate is made with respect to the uncertainty and variability associated with sampling data from a population.