Video: EG19M1-Statistics-Q7A

𝑋 is a normal random variable whose mean is zero and standard deviation is 𝜎. If 𝑝(𝑋 ≀ π‘˜πœŽ) = 0.877, find the value of π‘˜.

03:10

Video Transcript

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 π‘˜πœŽ is 0.877, find the value of π‘˜.

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, π‘˜πœŽ, is 0.877. And we want to work out the value of π‘˜.

First, we recall that a normal distribution is a bell-shaped curve symmetrical about its mean, πœ‡, 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 π‘˜πœŽ, 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 πœ‡ 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 πœ‡. So to work out the 𝑧-score for the value of π‘˜πœŽ in this distribution, we subtract the mean zero and then divide by the standard deviation 𝜎, giving π‘˜πœŽ minus zero over 𝜎. π‘˜πœŽ minus zero is just π‘˜πœŽ, and then dividing by 𝜎 gives π‘˜. So the 𝑧-score associated with a value of π‘˜πœŽ is just π‘˜.

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 π‘˜πœŽ will be π‘˜ lots of 𝜎, or π‘˜ 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 π‘˜.

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 π‘˜, 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.

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