To determine the percentage of pebbles that weigh more than 2.1 grams, follow these steps:
1. Identify Mean and Standard Deviation:
- Mean weight ([tex]\(\mu\)[/tex]): 2.6 grams
- Standard deviation ([tex]\(\sigma\)[/tex]): 0.4 grams
2. Find the Z-Score for the Given Weight (2.1 grams):
The Z-score formula is given by:
[tex]\[
Z = \frac{X - \mu}{\sigma}
\][/tex]
where [tex]\(X\)[/tex] is the given weight.
Substituting the known values:
[tex]\[
Z = \frac{2.1 - 2.6}{0.4} = \frac{-0.5}{0.4} = -1.25
\][/tex]
3. Use the Z-Score to Find the Corresponding Cumulative Probability:
Using the Z-table provided, we locate the value for [tex]\(Z = -1.25\)[/tex].
While the exact value isn't listed, we do know that:
- For [tex]\(Z = -1.2\)[/tex], the cumulative probability is approximately 0.11507.
- For [tex]\(Z = -1.3\)[/tex], the cumulative probability is approximately 0.09680.
Given that -1.25 falls between -1.2 and -1.3, the cumulative probability (area to the left of [tex]\(Z = -1.25\)[/tex]) can be interpolated:
[tex]\[
P(Z < -1.25) \approx 0.10565
\][/tex]
4. Convert the Cumulative Probability to the Desired Probability:
Since 0.10565 represents the probability that a pebble weighs less than 2.1 grams, we need to find the probability that a pebble weighs more than 2.1 grams:
[tex]\[
P(\text{weight} > 2.1) = 1 - P(Z < -1.25) = 1 - 0.10565 = 0.89435
\][/tex]
5. Convert the Probability to a Percentage:
[tex]\[
0.89435 \times 100 \approx 89.435\%
\][/tex]
6. Round to the Nearest Whole Percent:
The rounded percentage is:
[tex]\[
89\%
\][/tex]
Therefore, the percentage of pebbles that weigh more than 2.1 grams is 89%.
