the life expectancy of lung cancer patient treated with a new drug is normally distributed with a mean of 4 years and a standard deviation of 10 months what is the probability that a randomly selected lung cancer patient will last more than 5 years
Z = (x - μ)/SD (converted to years)
Z = (5 -4)/SD (converted to years)
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion higher than that Z score.
To solve this problem, we need to use the concept of standard deviation and the properties of a normal distribution. The given information tells us that the life expectancy of a lung cancer patient treated with a new drug follows a normal distribution with a mean of 4 years and a standard deviation of 10 months.
First, we need to convert the 5-year mark into the same units as the mean and standard deviation. Since the mean is given in years and the standard deviation is given in months, we need to convert 5 years into months. There are 12 months in a year, so 5 years is equal to 60 months.
Next, we need to calculate the Z-score (standardized score) for the 5-year mark using the formula:
Z = (X - mean) / standard deviation
where X is the value we are interested in (60 months) and mean is the mean value (4 years or 48 months). The standard deviation remains as 10 months.
Plugging in the values:
Z = (60 - 48) / 10
Z = 12 / 10
Z = 1.2
Now, we need to find the probability of a randomly selected lung cancer patient lasting more than 5 years, which is equivalent to finding the area under the normal distribution curve to the right of the Z-score of 1.2.
We can use a standard normal distribution table or a calculator to find this probability. If using a standard normal distribution table, we look for the Z-score of 1.2 and find the corresponding probability value. If using a calculator, we can use cumulative distribution functions (CDF) or the complementary cumulative distribution functions (CCDF).
Let's assume we are using a calculator. Using the CDF function, we calculate the probability of a Z-score being less than or equal to 1.2:
P(Z ≤ 1.2)
Then, to find the probability of a Z-score being greater than 1.2 (which is the probability of a randomly selected lung cancer patient lasting more than 5 years), we subtract the obtained probability from 1:
P(Z > 1.2) = 1 - P(Z ≤ 1.2)
Using the calculator or a standard normal distribution table, we can find the appropriate probability value and calculate the final probability.