𝑋 is a normal random variable whose mean is zero and standard deviation is 𝜎. If the probability that 𝑋 is less than or equal to 𝑘𝜎 is 0.877, find the value of 𝑘.
We’ve been told then that 𝑋 is a normal random variable with mean zero and standard deviation 𝜎. We can express this then using the usual notation for normal distribution. 𝑋 has a normal distribution with mean zero and standard deviation 𝜎. It, therefore, has a variance of 𝜎 squared. Now, we’re told that the probability that 𝑋 is less than or equal to some value, 𝑘𝜎, is 0.877. And we want to work out the value of 𝑘.
First, we recall that a normal distribution is a bell-shaped curve symmetrical about its mean, 𝜇, which in this question is zero. The area below the full curve is one. And the area to the left of any particular value gives the probability that 𝑋 is less than or equal to that value. In our case, the value is 𝑘𝜎, and the probability is 0.877.
Now, in order to work out these probabilities for a normal distribution, we use a 𝑧-score, which is found by subtracting the mean 𝜇 from that particular value 𝑋 and then dividing by the standard deviation 𝜎. This tells us how many standard deviations a particular value 𝑋 is from the mean 𝜇. So to work out the 𝑧-score for the value of 𝑘𝜎 in this distribution, we subtract the mean zero and then divide by the standard deviation 𝜎, giving 𝑘𝜎 minus zero over 𝜎. 𝑘𝜎 minus zero is just 𝑘𝜎, and then dividing by 𝜎 gives 𝑘. So the 𝑧-score associated with a value of 𝑘𝜎 is just 𝑘.
This make sense. Remember, a 𝑧-score tells us the number of standard deviations that a value is away from the mean. And if the mean is zero, then a value of 𝑘𝜎 will be 𝑘 lots of 𝜎, or 𝑘 standard deviations, above the mean. We would then use our standard normal distribution tables to work out the probability associated with a particular 𝑧-score. But, in this question, we’re going the other way. We know the probability 0.877. And we want to work backwards to find the corresponding 𝑧-score, which gives the value of 𝑘.
Our probability of 0.877 or 0.8770 is located here in the table. Looking across, we see that this is associated with a 𝑧-score of 1.10. And then looking upwards, we see that there is an additional 0.06. So we need to include a six in the second decimal place. So we find then that the 𝑧-score associated with a probability of 0.8770, and therefore the value of 𝑘, is 1.16. This means that for a normal random variable with a mean zero, the probability of that random variable 𝑋 taking a value less than or equal to 1.16 standard deviations above the mean is 0.877.