Questions: Find the necessary sample size. You wish to estimate the mean weight of machine components of a certain type and you require a 95% degree of confidence that the sample mean will be in error by no more than 0.009 g. Find the sample size required. A pilot study showed that the population standard deviation is estimated to be 0.02 g.

Find the necessary sample size. You wish to estimate the mean weight of machine components of a certain type and you require a 95% degree of confidence that the sample mean will be in error by no more than 0.009 g. Find the sample size required. A pilot study showed that the population standard deviation is estimated to be 0.02 g.

Transcript text: Find the necessary sample size. You wish to estimate the mean weight of machine components of a certain type and you require a $95 \%$ degree of confidence that the sample mean will be in error by no more than 0.009 g . Find the sample size required. A pilot study showed that the population standard deviation is estimated to be 0.02 g .

Solution

Solution Steps

Step 1: Determine the Z-Score

To find the necessary sample size, we first need to determine the Z-score corresponding to a $95\%$ confidence level. The Z-score can be calculated using the formula:

\[ Z = \text{PPF}\left(1 - \frac{1 - 0.95}{2}\right) = \text{PPF}(0.975) = 1.96 \]

Step 2: Calculate the Sample Size

Next, we use the Z-score, the population standard deviation ($\sigma$), and the desired margin of error to calculate the sample size using the formula:

\[ \text{Sample Size} = \left(\frac{Z \cdot \sigma}{\text{Margin of Error}}\right)^2 \]

Substituting the values:

\[ \text{Sample Size} = \left(\frac{1.96 \cdot 0.02}{0.009}\right)^2 \]

Calculating this gives:

\[ \text{Sample Size} = (1.96 \cdot 0.02 / 0.009)^2 = 18.9702 \approx 19.0 \]