Question

Samples of 1008 -hour shifts were randomly selected from the police records for each of two districts in a large city. The number of police emergency calls was recorded for each shift. The sample statistics are listed here: Find a \(90 \%\) confidence interval for the difference in the mean numbers of police emergency calls per shift between the two districts of the city. Interpret the interval.

Short answer

#tag_title# (Step 2: Determine the standard error of the difference) #tag_content# The standard error of the difference between the two means is calculated as: $$ SE(\bar{d}) = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} $$ where \(n_1\) and \(n_2\) are the sample sizes for district 1 and district 2, respectively. #tag_title# (Step 3: Find the critical value for a 90% confidence interval) #tag_content# To find the critical value, we can use a t-table or a t-distribution calculator. For a 90% confidence interval, the critical value is: $$ t_{\alpha/2} = t_{0.05} $$ #tag_title# (Step 4: Calculate the margin of error) #tag_content# The margin of error is calculated as: $$ ME = t_{0.05} \cdot SE(\bar{d}) $$ #tag_title# (Step 5: Calculate the confidence interval) #tag_content# The confidence interval is calculated as: $$ (\bar{d} - ME, \bar{d} + ME) $$ #tag_title# (Step 6: Interpret the interval) #tag_content# The 90% confidence interval for the difference in the mean numbers of police emergency calls per shift between the two districts means that we are 90% confident that the true difference in mean numbers of police emergency calls per shift lies within this interval. If the intervals contain positive values only, it suggests that the mean number of calls is higher in district 1; if it contains negative values only, it suggests that the mean number of calls is higher in district 2; if it contains both positive and negative values, it indicates that there is not enough evidence to conclude that one district has a higher mean number of calls than the other.

Step-by-step solution

(Step 1: Calculating the mean difference between the two districts)

Let \(\bar{x}_1\) be the mean number of emergency calls for district 1, \(\bar{x}_2\) be the mean number of emergency calls for district 2. Also let \(s_1^2\) be the sample variance for district 1 and \(s_2^2\) be the sample variance for district 2. The mean difference between the two districts is: $$ \bar{d} = \bar{x}_1 - \bar{x}_2 $$

Key concepts

Statistical Hypothesis Testing

Statistical hypothesis testing is a method used to determine if there is enough evidence to reject a null hypothesis about a population parameter based on sample data. In the context of our exercise involving police emergency calls, the null hypothesis might state that there is no difference in the mean numbers of calls between the two districts.

To test this hypothesis, statisticians use a test statistic that compares the observed sample statistic to the value stated in the null hypothesis. If this test statistic falls into a critical region, which is determined by the desired confidence level (in this case, 90%), the null hypothesis is rejected, suggesting that there may be a significant difference between the district means.

Understanding how to set up and interpret hypothesis tests is crucial for data analysis and making informed decisions based on that data. Correct application of hypothesis testing in various fields allows professionals to draw reliable conclusions from sample data.

Sample Statistics

Sample statistics are numerical values that summarize or describe features of a sample taken from a population. Examples include the sample mean, sample variance, and sample standard deviation. In the scenario provided, the sample statistics would include the mean number of police emergency calls per shift and the variances for each district.

These statistics are used as estimates of the corresponding population parameters they are meant to represent. For instance, the mean number of calls per shift, calculated from the sample data, provides an estimate of the average number of calls that might be expected across all shifts in the respective district's population.

Understanding how to calculate and interpret sample statistics is fundamental for making predictions about populations and is a cornerstone of statistical analysis.

Variance

Variance is a measure of how much the values in a dataset differ from the mean value. It's a crucial concept in statistics as it reflects the level of variability within a set of data. High variance indicates that the data points are spread out widely around the mean, and low variance indicates that they are clustered closely.

In our example with emergency calls, the variance for each district tells us how consistently the number of calls occurs around the mean for that district. When calculating a confidence interval for the mean difference between two independent samples, it is important to consider the variances because they affect the width of the confidence interval. If variances are high, the confidence interval will be wider, reflecting greater uncertainty about the mean difference.

Understanding variance is essential for interpreting the spread of data points and for performing more complex analyses, such as computing the standard error of the mean, which is used in forming confidence intervals.