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Go Tutorial Two machines are used to fill plastic bottles with dishwashing detergent. The standard deviations of fill volume are known to be \(\sigma_{1}=0.10\) fluid ounces and \(\sigma_{2}=0.15\) fluid ounces for the two machines, respectively. Two random samples of \(n_{1}=12\) bottles from machine 1 and \(n_{2}=10\) bottles from machine 2 are selected, and the sample mean fill volumes are \(\bar{x}_{1}=30.87\) fluid ounces and \(\bar{x}_{2}=30.68\) fluid ounces. Assume normality. (a) Construct a \(90 \%\) two-sided confidence interval on the mean difference in fill volume. Interpret this interval. (b) Construct a \(95 \%\) two-sided confidence interval on the mean difference in fill volume. Compare and comment on the width of this interval to the width of the interval in part (a). (c) Construct a \(95 \%\) upper-confidence interval on the mean difference in fill volume. Interpret this interval. (d) Test the hypothesis that both machines fill to the same mean volume, Use \(\alpha=0.05 .\) What is the \(P\) -value? (e) If the \(\beta\) -error of the test when the true difference in fill volume is 0.2 fluid ounces should not exceed 0.1 , what sample sizes must be used? Use \(\alpha=0.05\)

Step by step solution

01

Understand the Question

We need to find confidence intervals and perform hypothesis tests on the mean difference between two machines' fill volumes. The standard deviations, sample means, and sample sizes are provided. We will use these to calculate each result.

02

Calculate the Standard Error of the Mean Difference

The standard error of the mean difference is given by \(SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}\). Plugging in the values, we get \(SE = \sqrt{\frac{0.10^2}{12} + \frac{0.15^2}{10}}\). Calculate this to find the standard error.

03

Find Critical Values for the Confidence Intervals

For a 90% confidence interval, use the critical value \(z = 1.645\). For a 95% confidence interval, use \(z = 1.96\). These values correspond to a normal distribution.

04

Calculate the 90% Confidence Interval

Use the formula \((\bar{x}_1 - \bar{x}_2) \pm z \times SE\) to find the interval. Substitute \(\bar{x}_1 = 30.87\), \(\bar{x}_2 = 30.68\), and \(SE\) with the value calculated previously. Use \(z = 1.645\) for 90% confidence.

05

Calculate the 95% Confidence Interval

Repeat the process using a critical value of \(z = 1.96\) to obtain the 95% confidence interval.

06

Compare Widths of Confidence Intervals

Compare the width of the 90% confidence interval to the 95% confidence interval. The 95% interval will be wider because it provides more certainty.

07

Calculate the 95% Upper Confidence Interval

For an upper-confidence interval, use the formula \((\bar{x}_1 - \bar{x}_2) + z \times SE\), using \(z = 1.645\) for the upper bound. This gives the limit above which the true difference is unlikely to occur (95% confidence level).

08

Perform Hypothesis Test

To test if the means are equal, use the test statistic \(Z = \frac{(\bar{x}_1 - \bar{x}_2)}{SE}\) and compare to \(z = 1.96\) for a 5% significance. Compute the \(P\)-value and decide if it is less than \(\alpha = 0.05\).

09

Calculate Sample Size for Specific \(\beta\)-error

Given \(\beta = 0.1\), find \(n_1\) and \(n_2\) that satisfy this error for a true difference of 0.2 fluid ounces. Use the power of a test formula, considering the desired \(\alpha\) level and minimum detectable difference.

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Key Concepts

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

Confidence Intervals

Confidence intervals are vital in statistics as they provide a range within which we can say, with a certain level of confidence, that the true population parameter lies. This is particularly useful in hypothesis testing, allowing us to make inferences about the population based on sample data. For instance, when we constructed two-sided confidence intervals on the mean difference of fill volumes from two machines, we used sample means and standard deviations.
To calculate the confidence interval, we use the formula:

• For the 90% interval: \( CI_{90\%} = (\bar{x}_1 - \bar{x}_2) \pm z \times SE \)
• For the 95% interval: \( CI_{95\%} = (\bar{x}_1 - \bar{x}_2) \pm z \times SE \)

The confidence level denotes the percentage of all possible samples that can be expected to include the true population parameter. Thus, a narrower interval means more precision but less certainty, whereas a wider interval means more certainty but less precision. Confidence intervals help in understanding and interpreting statistical data effectively.

Sampling Distributions

In hypothesis testing, understanding the sampling distribution of the sample mean is crucial. A sampling distribution is the probability distribution of a statistic obtained through a large number of samples drawn from a specific population.
When assuming normality in our exercise, we recognize that the sample means from both machines form a distribution that is approximately normal. This assumption is pivotal because it simplifies calculations and allows for approximation using the standard normal distribution.
The key properties of sampling distributions include:

• The mean of the sampling distribution of the sample mean equals the population mean.
• The variance of the sampling distribution is lower than that of the population, scaled down by the sample size.

Hence, by knowing the sampling distribution, we can easily perform hypothesis tests and construct confidence intervals. Moreover, larger sample sizes tend to produce a distribution that is more normal and peaked, illustrating the Central Limit Theorem.

Error Analysis

Error analysis in statistics is essential for understanding the reliability of test results. Errors can often be classified into two types: Type I error, which occurs when the null hypothesis is wrongly rejected, and Type II error, when a null hypothesis that is false fails to be rejected.
In our exercise, when we tested the hypothesis that both machines fill to the same mean volume, we had to consider the significance level, \( \alpha = 0.05 \), which represents the probability of making a Type I error. Additionally, the \( \beta \)-error, or Type II error, is related to the power of the test, representing the probability of correctly rejecting a false null hypothesis.
By analyzing errors:

• We can determine the adequacy of sample sizes.
• We can balance the trade-offs between \( \alpha \) and \( \beta \); reducing one generally increases the other.

Reducing errors and understanding their implications helps improve the accuracy and validity of statistical conclusions.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the context of hypothesis testing or calculating confidence intervals, standard deviation plays a pivotal role in determining the standard error, which quantifies the uncertainty inherent in a sample estimate.
For our exercise involving the two machines, the standard deviations of fill volumes (\( \sigma_{1}=0.10 \) fluid ounces and \( \sigma_{2}=0.15 \) fluid ounces) were crucial in calculating the standard error of the mean difference. The formula used was:

• \( SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} \)

A smaller standard deviation means that data points tend to be close to the mean of the dataset, while a larger standard deviation indicates more spread out data points. Understanding standard deviation is fundamental to determine the confidence interval width and the precision of the sample estimate.

Sample Size Calculation

Sample size determination is critical when conducting hypothesis tests to ensure that results are statistically significant and reliable. Specifically, the sample size affects the margin of error and the power of a statistical test.
In our exercise, to control the risk of \( \beta \) error with a threshold not exceeding 0.1 when the true difference is 0.2 fluid ounces, it was vital to calculate appropriate sample sizes. The sample size influences both the accuracy of the estimate and the ability to detect an effect of a given magnitude with a specified level of confidence.
Key aspects of determining sample size:

• Higher sample sizes generally lead to more reliable and precise results.
• A balance is needed between practical constraints (e.g., cost and time) and the desired level of accuracy.
• Increased sample sizes reduce both \( \alpha \) and \( \beta \) errors, enhancing the test's power.

Careful calculation ensures that your study has sufficient power to detect meaningful effects without being inefficiently resource-heavy.