The distribution of the number of occurrences of the letter t on the pages of a book is found to be a normal distribution with a mean of 44 and a standard deviation of 18. If there are 500 pages in the book, which sentence most closely summarizes the data?
-The letter t occurs less than 26 times on approximately 170 of these pages.
-The letter t occurs less than 26 times on approximately 15 of these pages.
-The letter t occurs more than 26 times on approximately 420 of these pages.
-The letter t occurs more than 26 times on approximately 80 of these pages.

Question

The distribution of the number of occurrences of the letter t on the pages of a book is found to be a normal distribution with a mean of 44 and a standard deviation of 18. If there are 500 pages in the book, which sentence most closely summarizes the data?
-The letter t occurs less than 26 times on approximately 170 of these pages.
-The letter t occurs less than 26 times on approximately 15 of these pages.
-The letter t occurs more than 26 times on approximately 420 of these pages.
-The letter t occurs more than 26 times on approximately 80 of these pages.

anonymous · Answer

Help.!!

anonymous · Answer

Compute the probability that the letter occurs more less than 26 times, given by $$P(X<26)$$ where $X$ denotes the random variable for the number of times the letter t shows up. Use the variable transformation $Z=\dfrac{X-\mu}{\sigma}$ to convert to the standard normal distribution, where $\mu$ is the mean and $\sigma$ is the standard deviation. $$P(X<26)=P\left(\frac{X-44}{18}<\frac{26-44}{18}\right)=P(Z<-1)$$ Refer to a table to determine the probability, or use the rule here: http://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule The probability $P(Z<-1)$ is the proportion of the population that lies at least one standard deviation to the left of the mean: |dw:1420313106793:dw|