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A consumer organization tested two paper shredders, the Piranha and the Crocodile, designed for home use. Each of 10 randomly selected volunteers shredded 100 sheets of paper with the Piranha, and then another sample of 10 randomly selected volunteers each shredded 100 shects with the Crocodile. The Piranha took an average of 203 seconds to shred 100 sheets with a standard deviation of 6 seconds. The Crocodile took an average of 187 seconds to shred 100 sheets with a standard deviation of 5 seconds. Assume that the shredding times for both machines are normally distributed with equal but unknown standard deviations. a. Construct a \(99 \%\) confidence interval for the difference between the two population means. b. Using a \(1 \%\) significance level, can you conclude that the mean time taken by the Piranha to shred 100 sheets is higher than that for the Crocodile? c. What would your decision be in part \(\mathrm{b}\) if the probability of making a Type 1 error were zero? Explain.

Step by step solution

01

Construct a 99% Confidence Interval

Calculate the pooled standard deviation using the formula \[\sqrt{[(n1-1) * s1^2 + (n2-1) * s2^2]/(n1+n2-2)}\] where n1, n2 are the sizes of the samples (10 in our case) and s1, s2 are their standard deviations (6 and 5 seconds respectively here). This gives the pooled standard deviation. Create the confidence interval using the formula \[ X¯1 - X¯2 ± Z∗√[ σ1^2/n1 + σ2^2/n2 ]\] where Z is the Z-score for a 99% confidence interval (approximate value around 2.58), X¯1, X¯2 are the sample means (203 and 187 seconds), σ1, σ2 are their standard deviations.

02

1% Significance Level Hypothesis Test

Formulate the null hypothesis (H0: μ1-μ2=0) that there is no difference in the mean shredding times, and the alternative hypothesis (H1: μ1-μ2>0) that Piranha's mean time is greater. Calculate the test statistic (t) using \[t = (X¯1 - X¯2 - D) / √((σ1^2/n1) + (σ2^2/n2))\] where D is the difference in population means under the null hypothesis (0 here). Compare the calculated t with the critical t-value for a 1% significance level to accept or reject the null hypothesis.

03

Risk of Type 1 Error

A Type 1 error is the incorrect rejection of a true null hypothesis, meaning here it would be judging Piranha to be slower when it is not. If the probability of this error is 0, the null hypothesis cannot be rejected upon calculation, therefore the decision from step 2 could potentially change.

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Key Concepts

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

Understanding Confidence Interval

A confidence interval offers a range of values within which we expect the true population parameter to lie. In this exercise, we deal with a 99% confidence interval, which means we are 99% sure that the true difference between the mean shredding times of the Piranha and the Crocodile lies inside the calculated interval.
The confidence interval helps to understand how much overlap there is between the two shredders' performances. Using the means and standard deviations provided, we first calculate a pooled standard deviation. This pooled value helps create a more refined estimate of the common standard deviation across both machines.
Next, we use the formula for the confidence interval \[ X¯1 - X¯2 \pm Z^* \sqrt{\left( \frac{\sigma_1^2}{n1} + \frac{\sigma_2^2}{n2} \right)} \]where \(Z\) is the Z-score corresponding to the chosen confidence level, in this case, approximately 2.58 for 99%. This approach allows us to quantify our uncertainty around the measured differences in performance.

Comprehending Type 1 Error

A Type 1 error occurs when we incorrectly reject a true null hypothesis. In this context, it would mean concluding that the Piranha is indeed slower than the Crocodile, when in fact, there's no true difference in their mean shredding times.
The consequence of a Type 1 error could lead to wrong business or consumer choices, like discontinuing a product based on incorrect data analysis. Mitigating Type 1 error involves setting a conservative significance level, such as 1% in this case of hypothesis testing.
By using a 1% significance level, the probability of committing a Type 1 error is reduced to 1%, adding robustness to our decision-making process. Understanding this error type is essential in hypothesis testing to ensure reliable results and conclusions.

Significance Level: Key in Hypothesis Testing

The significance level, denoted as \( \alpha \), is a threshold we set to decide whether to reject the null hypothesis. In this problem, we utilize a 1% significance level, which means there's a 1% risk of making a Type 1 error—rejecting a true null hypothesis.
This level dictates how extreme the sample needs to be for us to be confident there's a real effect, like the Piranha taking more time. By calculating the test statistic and comparing it with the critical value for this level, we determine whether the observed difference is significant enough to reject the null hypothesis.
Setting a low significance level makes it more challenging to prove a difference exists unless there is strong evidence, thus ensuring only robust and clear differences influence our conclusions. This careful approach helps safeguard against unintentional errors and enhances the reliability of the results.

Exploring Standard Deviation

Standard deviation measures the amount of variation or dispersion in a set of values. It's central to understanding data spread and variability, crucial in the exercise where we're comparing two sample means with different shredders.
In this case, smaller standard deviations for the Piranha (6 seconds) and the Crocodile (5 seconds) indicate their shredding times are close to their mean values, implying consistency and reliability in performance.
The pooled standard deviation is used to calculate the confidence interval and the test statistic in hypothesis testing. It provides a more unified measure of variability across both samples. This parameter facilitates the estimation of the mean differences' confidence intervals and hypothesis tests.
Overall, standard deviation is key to understanding the nature of variability and plays a major role in determining the reliability and consistency of statistical conclusions across datasets.