3.The lifespan of 300 Giant turtles is found to follow a Normal distribution ~ N(70,144).

(a)What is the standard error for this sample’s mean? (b) What is the probability of a turtle living more than 85 years?
(c)What is the probability of a turtle living between 60 and 65 years?
(d)Find the 98% confidence interval for the mean lifetime.
(e) Below what age do the shortest lived 15% of turtles die?

SEm = SD/√n

Z = (score-mean)/SD or SEm, depending on whether you are talking about a distribution of individual scores or means respectively.

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.

(a) The standard error for the sample mean can be calculated using the formula:

Standard Error = Standard Deviation / √(Sample Size)

In this case, the standard deviation is given as 144, and the sample size is not provided. Without the sample size, we cannot calculate the standard error. Please provide the sample size in order to calculate the standard error.

(b) To find the probability of a turtle living more than 85 years, we need to calculate the z-score for 85 and find the area under the standard normal distribution curve for values greater than that z-score.

We can use the formula:
z = (x - μ) / σ

In this case, x = 85, μ = 70, and σ = √144 = 12 (standard deviation).

z = (85 - 70) / 12 = 1.25

To find the probability, we can look up the z-score in a standard normal distribution table or use a calculator. The probability of a turtle living more than 85 years is the area under the curve to the right of the z-score.

(c) To find the probability of a turtle living between 60 and 65 years, we need to calculate the z-scores for 60 and 65 and find the area under the curve between those z-scores.

Using the same formula as in part (b) and the given values, we calculate the z-scores:

z1 = (60 - 70) / 12 = -0.83
z2 = (65 - 70) / 12 = -0.42

To find the probability, we can look up the z-scores in a standard normal distribution table or use a calculator. The probability of a turtle living between 60 and 65 years is the area under the curve between z1 and z2.

(d) To find the 98% confidence interval for the mean lifetime, we need to find the z-score corresponding to a 98% confidence level. The z-score can be found using a standard normal distribution table or a calculator.

The formula to calculate the confidence interval is:
Confidence Interval = Sample Mean ± (z * Standard Error)

Since the sample size is not provided, we cannot calculate the standard error and consequently the confidence interval. Please provide the sample size to calculate the confidence interval.

(e) To find the age below which the shortest-lived 15% of turtles die, we need to find the z-score corresponding to the 15th percentile.

Using a standard normal distribution table or a calculator, we can find the z-score corresponding to the 15th percentile. Let's denote it as z15.

Using the formula:
z15 = invNorm(0.15)

Once we have the z15 value, we can use it to find the corresponding age using the formula:
Age = (z * Standard Deviation) + Mean

In this case, the mean is given as 70 and the standard deviation is given as 144. Substituting these values, we can find the age below which the shortest-lived 15% of turtles die.

Note: invNorm() refers to the inverse normal distribution function, which can be found on calculators or statistical software.