Question

1. Jarrid Medical, Inc., is developing a compact kidney dialysis machine, but the company's chief engineer, Mike Crowe, is having trouble controlling the variability in how fast the fluid moves through the kidney. the device. Medical standards require the flow per hour to be 4.25 liters, 98% of the time. Mr. Crowe performs tests on the prototype with the following results for each hourly flow test performed: 4.17 4.32 4.21 4.22 4.29 4.19 4.29 4.34 4.33 4.22 4.28 4.33 4.52 4.29 4.43 4.39 4.44 4.34 4.3 4.41 With the information obtained from the sample, it is necessary to find the confidence interval that reflects the expected flow per hour in 98% of the time of the developed compact machine and determine if it is satisfying the medical standards. Does the prototype satisfy medical standards? Present all the steps required to construct your confidence interval and an analysis that justifies your answer to the established question.

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Step 1: Calculate the sample mean ( \bar{x} ) and the sample standard deviation ( s ).

Given sample data:

4.17, 4.32, 4.21, 4.22, 4.29, 4.19, 4.29, 4.34, 4.33, 4.22, 4.28, 4.33, 4.52, 4.29, 4.43, 4.39, 4.44, 4.34, 4.3, 4.41

\bar{x} = \frac{4.17 + 4.32 + 4.21 + 4.22 + 4.29 + 4.19 + 4.29 + 4.34 + 4.33 + 4.22 + 4.28 + 4.33 + 4.52 + 4.29 + 4.43 + 4.39 + 4.44 + 4.34 + 4.3 + 4.41}{20} = 4.281

s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}} = \sqrt{\frac{(4.17-4.281)^2 + (4.32-4.281)^2 + \cdots + (4.41-4.281)^2}{19}} = 0.086

Step 2: Determine the t-score for a 98% confidence level withn-1 = 19 degrees of freedom. From the t-distribution table:

t_{\alpha/2,19} \approx 2.539

Step 3: Calculate the margin of error (ME).

ME = t_{\alpha/2,19} \cdot \frac{s}{\sqrt{n}} = 2.539 \cdot \frac{0.086}{\sqrt{20}} = 0.049

Step 4: Construct the confidence interval.

\bar{x} \pm ME = 4.281 \pm 0.049 \Rightarrow (4.244, 4.319)

Step 5: Analyze if the mean flow rate meets the medical standards.

The required mean flow rate is 4.25 liters. The confidence interval (4.244, 4.319) includes 4.25, thus the prototype does satisfy the medical standards for the mean flow rate 98% of the time.

\text{Confidence interval: } (4.244, 4.319)

\text{Satisfies medical standards: Yes}

Given sample data:

Step 2: Determine the t-score for a 98% confidence level with

Step 3: Calculate the margin of error (ME).

Step 4: Construct the confidence interval.

Step 5: Analyze if the mean flow rate meets the medical standards.

The required mean flow rate is 4.25 liters. The confidence interval (4.244, 4.319) includes 4.25, thus the prototype does satisfy the medical standards for the mean flow rate 98% of the time.

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