Question

What would be the required number of readings if the analyst wanted to be 87 percent confident that the mean observed time was within ± 5 percent of the true mean and he established the following values for an element after observing 19 cycles: 0.09,0.08,0.10,0.12,0.09,0.08,0.09,0.12,0.11,0.12,0.09,0.10,0.12,0.10,0.08,0.09,0.10, 0.12,0.09 ?

Answer 1

To calculate the required number of readings, use the formula n = (Z * σ / E)^2, where Z is the Z-score, σ is the sample standard deviation, and E is the desired margin of error as a decimal. First, calculate the standard deviation using the given data, and then use the formula to determine the required number of readings.

Step-by-step explanation:

To calculate the required number of readings, we need to use the formula:

n = (Z * σ / E)^2

where:

1. n is the required number of readings
2. Z is the Z-score corresponding to the desired confidence level. For 87% confidence level, the Z-score is approximately 1.15.
3. σ is the standard deviation of the sample
4. E is the desired margin of error as a decimal (5% as 0.05)

In this case, we need to calculate the standard deviation σ using the given data:

Step 1: Calculate the sample mean (x-bar) by summing all the readings and dividing by the number of readings (19 in this case).

Step 2: Subtract the sample mean from each reading and square the result.

Step 3: Sum up all the squared differences from Step 2.

Step 4: Divide the result from Step 3 by the number of readings (19 - 1) to get the sample variance (s^2).

Step 5: Take the square root of the sample variance to get the standard deviation (s).

Once we have the standard deviation σ, we can plug in the values into the formula to calculate the required number of readings.