Using the z-scores formula and the invNorm function, calculate z-scores for the lower and upper limits, find corresponding percentiles, and subtract to determine the probability that the sample mean is between 2 and 3 hours.
To find the probability that the sample mean is between two hours and three hours, we need to find the z-scores corresponding to these values and then use the normal distribution table.
Given:
- Mean
: 2.5 hours
- Standard Deviation
: 0.25 hours
- Sample Size (\(n\)): 60
- Lower Limit: 2 hours
- Upper Limit: 3 hours
First, find the z-scores for both limits using the formula:
![\[ z = \frac{{X - \mu}}{{(\sigma)/(√(n))}} \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/o3bqwcejm4evq0w5vglfw54izu3bmz0qiu.png)
For the lower limit (2 hours):
![\[ z_{\text{lower}} = \frac{{2 - 2.5}}{{(0.25)/(√(60))}} \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/ckjs7q6dfameuw94ljd6orybqxukh3w894.png)
For the upper limit (3 hours):
![\[ z_{\text{upper}} = \frac{{3 - 2.5}}{{(0.25)/(√(60))}} \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/1hw1iadx1e85k6fn092x1n8n0bgkdc7bpq.png)
Now, use these z-scores to find the corresponding percentiles using the invNorm function on the calculator:
![\[ P(\text{Lower Limit}) = \text{invNorm}(z_{\text{lower}}, 0, 1) \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/i6r66d9ynsihmawseq4lhfxqo258jwbgwb.png)
![\[ P(\text{Upper Limit}) = \text{invNorm}(z_{\text{upper}}, 0, 1) \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/aldeor8wdrus5prbh89d2a886ou885s7ol.png)
Finally, the probability that the sample mean is between two hours and three hours is given by the difference of these percentiles:
![\[ P(2 < \text{Sample Mean} < 3) = P(\text{Upper Limit}) - P(\text{Lower Limit}) \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/82qmafahw57j5dmuqwombq8dqd8suttgmh.png)