Question

A survey is taken of the citizens in a city as to whether or not they support the zoning plan that the city council is considering. The responses are: Yes, No, Indifferent, and Otherwise. Let \(p_{1}, p_{2}, p_{3}\), and \(p_{4}\) denote the respective true probabilities of these responses. The results of the survey are: $$ \begin{array}{|c|c|c|c|} \hline \text { Yes } & \text { No } & \text { Indifferent } & \text { Otherwise } \\ \hline 60 & 45 & 70 & 25 \\ \hline \end{array} $$ (a) Obtain the mles of \(p_{i}, i=1, \ldots, 4\). (b) Obtain \(95 \%\) confidence intervals, \((4.2 .7)\), for \(p_{i}, i=1, \ldots, 4\).

Short answer

The MLEs are \(p_{1} = 0.3\), \(p_{2} = 0.225\), \(p_{3} = 0.35\), and \(p_{4} = 0.125\). The 95% confidence intervals would be calculated using the respective MLE and SE for each response.

Step-by-step solution

Calculation of the MLE for each response

To find the maximum likelihood estimates (MLE) for each response, sum up all responses which comes to 200. Then, divide the number of each response by the total. Hence, the MLE for 'Yes' \(p_{1}\) is \(60/200 = 0.3\), for 'No' \(p_{2}\) it is \(45/200 = 0.225\), for 'Indifferent' \(p_{3}\) it is \(70/200 = 0.35\) and for 'Otherwise' \(p_{4}\) it is \(25/200 = 0.125\).

Calculation of the standard errors

Next, we calculate the standard error (SE) for each MLE using the formula \[SE = \sqrt{\frac{p(1-p)}{n}}\]. For 'Yes', \[SE_1 = \sqrt{\frac{0.3 \times 0.7}{200}}\]. For 'No', \[SE_2 = \sqrt{\frac{0.225 \times 0.775}{200}}\]. For 'Indifferent', \[SE_3 = \sqrt{\frac{0.35 \times 0.65}{200}}\]. And for 'Otherwise', \[SE_4 = \sqrt{\frac{0.125 \times 0.875}{200}}\]. Calculate these to get the SE for every response.

Calculation of the Confidence Intervals

To get 95% confidence intervals, we use the formula \[CI = p \pm 1.96 \times SE\]. For each \(p_i\), calculate the confidence interval in this way. For example, the confidence interval for 'Yes' would be calculated as \(CI_1 = 0.3 \pm 1.96 \times SE_1\) . Repeat this process for \(p_2\), \(p_3\), and \(p_4\) to find all confidence intervals.

Key concepts

Confidence Interval Calculation

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter. The purpose of calculating a confidence interval is to give an estimate about where the true population parameter lies, with a certain degree of assurance called the confidence level, typically expressed as a percentage, such as 95%.

When we report that we are 95% confident that the true value of the parameter lies within the confidence interval, it doesn't mean that there is a 95% probability that the parameter is within the interval; instead, it implies that if we were to take many samples and build a confidence interval from each of them, approximately 95% of those intervals would contain the true parameter value.

To calculate the confidence interval for the proportion, as seen in the survey example, we used the formula: \[\begin{equation} CI = \bar{x} \[6pt\bar{x} \pm Z \times SE \] \end{equation}\] where \bar{x} is the sample mean, Z is the Z-value from the standard normal distribution for the desired confidence level (1.96 for 95%), and SE is the standard error of the estimate. For the proportions in the survey problem, the use of the standard error incorporates the variability inherent in the estimate due to the size of the sample, allowing us to construct an interval that the population parameter is likely to fall in.

Probability Theory

Probability theory is the branch of mathematics that deals with the likelihood of different outcomes in various scenarios. At the core, a probability is a value between 0 and 1, inclusive, where 0 indicates that an event will not occur, and 1 implies that it is certain. When calculating probabilities for different types of events, mathematicians and statisticians employ a variety of principles and properties, such as independence, mutual exclusivity, and the addition and multiplication rules.

In the context of the survey problem we analyzed, each response can be seen as a distinct event with a certain probability of occurring. Since these events are mutually exclusive (a single response can't be 'Yes' and 'No' at the same time), the sum of their probabilities must equal 1. This principle is used in maximum likelihood estimation to determine the most probable set of outcomes given the observed data. Understanding these basic concepts of probability is essential for making sense of data and making informed decisions based on statistical analysis.

Statistical Hypothesis Testing

Statistical hypothesis testing is a method of making decisions using data, whether from a controlled experiment or an observational study (as in the survey example). In hypothesis testing, two opposing hypotheses about a population parameter are considered: the null hypothesis (\[\begin{equation} H_{0} \end{equation}\] ) and the alternative hypothesis (\[\begin{equation} H_{1} \or \H_{a} \end{equation}\] ). The null hypothesis is typically a statement of 'no effect' or 'no difference,' and the alternative is what the researcher aims to support.

Statistical tests produce a p-value, which measures the strength of the evidence against the null hypothesis. This p-value helps in deciding whether to reject the null hypothesis or not, based on a predefined significance level (\[\begin{equation} \alpha \end{equation}\] ), usually set at 0.05 or 5%. If the p-value is less than \[\begin{equation} \alpha \end{equation}\] , the null hypothesis is rejected in favor of the alternative.

In relation to confidence intervals, if a 95% confidence interval does not contain the value of the parameter specified by the null hypothesis, it suggests that at a significance level of 5%, the null hypothesis can be rejected. This adds another layer of understanding to the reliability of the estimated parameters obtained through the MLE technique used in the survey’s analysis.