the average numbers of hours a cat sleep is approximately normally distributed with a mean of 15 hours and a standard deviation of 1.7 hours; what percentage of cats sleep between 13 and 18 hours
z = (13-15)/1.7
z = -1.18
z = (18-15)/1.7
z = 1.76
p (-1.18 < z < 1.76) = 0.9608-0.1190 = 0.8418
84.18%
To find the percentage of cats that sleep between 13 and 18 hours, we need to calculate the area under the normal distribution curve between these two values.
First, let's convert these values to their corresponding z-scores. The z-score measures the number of standard deviations an observation is from the mean in a normal distribution.
The z-score can be calculated using the formula:
z = (x - μ) / σ
where:
- x is the observed value (in this case, the hours of sleep)
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
For 13 hours:
z₁ = (13 - 15) / 1.7 = -1.18
For 18 hours:
z₂ = (18 - 15) / 1.7 = 1.76
Next, we need to determine the area under the normal curve between these two z-scores. We can do this using a standard normal distribution table or by utilizing statistical software.
If using a standard normal distribution table, we can look up the probability associated with each z-score and find the difference between these two probabilities. The cumulative distribution function (CDF) is used to calculate the probability of observing a value less than or equal to a given z-score.
Alternatively, statistical software like Python or R can provide more accurate results by directly calculating the areas under the curve. For example, using Python with the scipy.stats module:
```python
from scipy.stats import norm
prob_between = norm.cdf(1.76) - norm.cdf(-1.18)
percentage_between = prob_between * 100
```
This code calculates the probability (or percentage) between the two z-scores using the cumulative distribution function (CDF) from the normal distribution in the scipy.stats module.
In this case, the percentage of cats that sleep between 13 and 18 hours would be calculated as the result of the above code snippet.