Hello I've been battling with this particular question from my statistics textbook for hours. Can someone kindly help with this.
Note: it is not an assignment question. I'm solving all questions in the textbook.
Thanks.
Suppose a professor is carrying out a statistical research in New York and found out that the duration of marriages without the consent of parents have been found to be normally distributed with mean of 10 years and standard deviation of 3 years.
a. Find the probablity that a single marriage of this nature would last longer than 18 years.
b. What is the probability that the mean of a simple random sample of 50 such marriages would not last longer than 9 years?
c. The probability is 0.80 that the mean of a random sample of 80 such marriages is between X₁ and X₂. Find X₁ and X₂ using symmetrical limits about the mean
For part a), all you need to do is integrate the normal distribution curve (with the specified mean and standard deviation) over the interval $(18,\infty)$ using the error function, technology, or even by standardizing the data and using tables.
Part b) and part c) both rely on the fact that a simple random sample of means from a normal distribution will also be normally distributed with the same mean $\mu$ but rather a standard deviation of $\frac{\sigma}{\sqrt{n}}$.
Thus for b) you can integrate the normal curve (using the modified standard deviation) over the interval $(-\infty,9)$ to obtain the required probability.
In c), you can use the fact of symmetrical limits to solve the equation $\int_{\mu-k}^{\mu+k} f(x)\,dx=0.8$ for $k$, where $f(x)$ is the normal curve with your mean and modified standard deviation. In this case, $\mu-k$ would represent $X_1$ and $\mu+k$ would represent $X_2$.