Use your TI83 (or Excel):
A certain giant tortoise has a life span that is normally distributed with a mean age of 75 years and a standard deviation of 17 years. Determine the probability that a random sample of 39 such tortoises has an average life span between 73 and 78 years.
Round to four decimal places.
Answer
Final Answer: The probability that a random sample of 39 such tortoises has an average life span between 73 and 78 years is approximately \(\boxed{0.6335}\).
Step 1 ：The problem is asking for the probability that the average life span of a sample of 39 tortoises is between 73 and 78 years. This is a problem of normal distribution. The mean and standard deviation of the population are given, and we need to find the probability for the sample mean.
Step 2 ：The formula for the standard deviation of the sample mean (also known as the standard error) is the standard deviation of the population divided by the square root of the sample size. In this case, the standard deviation of the population is 17 years, and the sample size is 39 tortoises. So, the standard error is \( \frac{17}{\sqrt{39}} \approx 2.722 \) years.
Step 3 ：We can then convert the sample mean range (73 to 78) to z-scores, which are standard scores that tell us how many standard deviations away from the mean a value is. The formula for a z-score is \( \frac{value - mean}{standard deviation} \). So, the z-score for 73 is \( \frac{73 - 75}{2.722} \approx -0.735 \) and the z-score for 78 is \( \frac{78 - 75}{2.722} \approx 1.102 \).
Step 4 ：Finally, we can use the z-scores to find the probability that the sample mean is within the given range. This can be done using the cumulative distribution function (CDF) for a standard normal distribution, which gives the probability that a random variable is less than or equal to a given value. The probability that the sample mean is between 73 and 78 is the probability that the z-score is less than or equal to the z-score for 78, minus the probability that the z-score is less than or equal to the z-score for 73. So, the probability is \( 0.865 - 0.231 \approx 0.634 \).
Step 5 ：Final Answer: The probability that a random sample of 39 such tortoises has an average life span between 73 and 78 years is approximately \(\boxed{0.6335}\).
