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A town that recently started a single-stream recycling program provided 60-gallon recycling bins to 25 randomly selected households and 75-gallon recycling bins to 22 randomly selected households. The total volume of recycling over a 10-week period was measured for each of the households. The average total volumes were 382 and 415 gallons for the households with the 60 - and 75 -gallon bins, respectively. The sample standard deviations were \(52.5\) and \(43.8\) gallons, respectively. Assume that the 10 -week total volumes of recycling are approximately normally distributed for both groups and that the population standard deviations are equal. a. Construct a \(98 \%\) confidence interval for the difference in the mean volumes of 10 -week recycling for the households with the 60 - and 75 -gallon bins. b. Using the \(2 \%\) significance level, can you conclude that the average 10 -week recycling volume of all households having 60 -gallon containers is different from the average volume of all households that have 75 -gallon containers?

Step by step solution

01

Identify the given quantities

The sample sizes, means, and standard deviations for both groups are given as: For 60-gallon bins: \(n_1=25\), \(\overline{x}_1=382\), \(s_1=52.5\), and for 75-gallon bins: \(n_2=22\), \(\overline{x}_2=415\), \(s_2=43.8\).

02

Calculate the standard error (SE) for the sampling distribution of differences in means

SE is calculated using the formula: \(SE=\sqrt{\left(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\right)}\). Substituting given values into the formula, calculates the SE of the sampling distribution of the difference.

03

Construct the 98% confidence interval

The formula for a confidence interval is: \((\overline{x}_1-\overline{x}_2)\pm (t_{crit} \times SE)\). In this case, since the sample sizes are less than 30, we use the t-critical value (t-crit). The t-crit can be determined using a t-table (for the 98% confidence and degrees of freedom being \( n_1 + n_2 - 2\)).

04

Test the hypothesis using the 2% significance level

To test if the means are different, we calculate t using the formula: \(t=\frac{\overline{x}_1-\overline{x}_2}{SE}\). Then, we compare this t with t-critical for two-tailed test at 2% significance level. If our calculated t is greater than t-crit, we reject the null hypothesis and conclude the means are different. Otherwise, we fail to reject the null hypothesis and can't conclude they are different.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Recycling Program

In the exercise, a town implemented a single-stream recycling program with two different bin sizes given to randomly selected households. This is a practical application of recycling efforts that promote environmental sustainability. Single-stream recycling is convenient because households do not need to sort recyclables, making participation easier and potentially increasing recycling rates. This experiment examines the average recycling volume over ten weeks for homes with different bin sizes.
By comparing the average volumes for the 60-gallon and 75-gallon bins, researchers assess whether the size of the recycling bin influences how much each household recycles. This can reveal if larger bins encourage more recycling simply because they can hold more, or if the convenience does not impact recycling habits significantly. With data-driven decisions, such strategies can be refined and optimized for better environmental outcomes.

Sampling Distribution

The concept of the sampling distribution is crucial when analyzing how sample data reflects larger populations. In this context, we have two sample groups: households with 60-gallon bins and those with 75-gallon bins. Each group represents a random sample intended to mirror the entire population of households in the recycling program.
When we take a sample from a population, the average of the sample (sample mean) can provide an estimate of the population mean. If multiple samples are drawn, the distribution of these sample means forms what is known as the sampling distribution. The central idea is that with enough samples, the sampling distribution approximates a normal distribution, thanks to the Central Limit Theorem. This is essential when constructing confidence intervals and performing hypothesis tests, as it helps predict the variability of sample means and thus how they might compare to a true population mean.

Standard Error

Standard error (SE) measures the variability or spread of sample statistics, such as the mean, from their true population values. In this recycling study, it helps quantify the uncertainty in the difference between the average volumes recycled by the two groups.
The standard error of the difference in means is particularly important here. It's calculated using the formula:\[SE = \sqrt{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)}\]where \(s_1\) and \(s_2\) are the sample standard deviations, and \(n_1\) and \(n_2\) are the sample sizes for each group. A smaller SE indicates less variability in the sample mean estimates, suggesting they are likely close to the true population means. Understanding this helps in making more precise confidence intervals and conducting more reliable hypothesis tests, as seen in the exercise.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data. In this exercise, it helps determine if the average recycling volumes differ between households with 60-gallon and 75-gallon bins.
The null hypothesis typically states that there is no difference in recycling volumes between the two groups—any observed difference is due to random chance. The alternative hypothesis suggests there is a genuine difference. Using the 2% significance level means we are looking for a strong enough difference—our confidence in this conclusion must be very high. In statistical terms, we calculate the test statistic:\[t = \frac{\overline{x}_1 - \overline{x}_2}{SE}\]where \(\overline{x}_1\) and \(\overline{x}_2\) are the means, and SE is the standard error of the difference. This test statistic is compared against a critical value from the t-distribution. If the test statistic exceeds this critical value, we reject the null hypothesis, indicating the means are likely different.