Now that we've had practice calculating $z$z-scores and know how to interpret them, we can use $z$z-scores to identify events that are more or less likely to occur. This is because $z$z-scores measure how far away a score is from the mean relative to the rest of the data set, and the further a value is from the mean, the less likely it is to occur.
Exploration
The arrival times for a particular train are approximately normally distributed with an expected arrival time of $9:00$9:00 am and a standard deviation of $10$10 minutes. The bell curve looks like this:
So for the arrival time of $8:50$8:50 am, the corresponding $z$z-score is $-1$−1 since this arrival time is $1$1 standard deviation below the mean. An arrival time of $9:10$9:10 am is $1$1 standard deviation above the mean, so its $z$z-score is $1$1. Following the above procedure, we can obtain the following table:
| Times | $8:30$8:30 am | $8:40$8:40 am | $8:50$8:50 am | $9:00$9:00 am | $9:10$9:10 am | $9:20$9:20 am | $9:30$9:30 am |
|---|---|---|---|---|---|---|---|
| $z$z-scores | $-3$−3 | $-2$−2 | $-1$−1 | $0$0 | $1$1 | $2$2 | $3$3 |
We might be interested in answering some of the following questions:
- If I’m waiting for the train is it more likely that it arrives after $9:00$9:00 am or after $9:10$9:10 am?
- Is it more likely that the train will arrive between $8:40$8:40 am and $8:50$8:50 am or $9:00$9:00 am and $9:10$9:10 am?
For each of these cases we will calculate and interpret their $z$z-scores.
1. A time of $9:00$9:00 am has a $z$z-score of $0$0 while a time of $9:10$9:10 am has a $z$z-score of $1$1. So for a train to arrive after $9:00$9:00 am, they have to fall in this region of the bell curve:
For a train to arrive after $9:10$9:10 am, they have to fall in this region of the bell curve:
As you can see, it’s more likely that the train will arrive after $9:00$9:00 am, because more of the bell curve is shaded. We can go even one step further to say that $50%$50% of the time, the train will arrive after $9:00$9:00 am since the normal distribution is symmetrical.
Using the empirical rule, we know that $68%$68% of arrivals are within $1$1 standard deviation of the mean. This means $32%$32% of arrivals are outside $1$1 standard deviation of the mean, which we can see below.
The curve is symmetrical so $16%$16% of arrivals are before $8:50$8:50 am and $16%$16% of arrivals are after $9:10$9:10 am.
In summary, $50%$50% of the time the train arrives after $9:00$9:00 am, and $16%$16% of the time the train arrives after $9:10$9:10 am.
2. Using the $z$z-scores, a train will arrive between $8:40$8:40 am and $8:50$8:50 am if it's between $1$1 and $two$two standard deviations below the mean.
Using the empirical rule, we know the percentage of trains that arrive in the following two regions.
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 |
We can then subtract one percentage from the other to find that $13.5%$13.5% of trains arrive between $8:40$8:40 am and $8:50$8:50 am. By symmetry, we can see that $34%$34% of trains arrive between $9:00$9:00 am and $9:10$9:10 am, so this time range is more than twice as likely.
Practice questions
question 1
The amount of time spent waiting in the reception area at a doctor's office is approximately normally distributed.
The $z$z-scores of the waiting times of four patients, represented by the letters $K$K, $L$L, $M$M, and $N$N are given in the table below.
| Patient | $K$K | $L$L | $M$M | $N$N |
|---|---|---|---|---|
| $z$z-score | $-2.91$−2.91 | $1.84$1.84 | $-2.15$−2.15 | $1.48$1.48 |
Which event is least likely?
Waiting longer than $L$L.
AWaiting longer than $M$M.
BWaiting longer than $N$N.
CWaiting longer than $K$K.
DWhich event is more likely?
Waiting longer than $M$M but less than $N$N.
AWaiting longer than $K$K but less than $L$L.
BWhich event is more likely?
Waiting less than $M$M or longer than $N$N.
AWaiting less than $K$K or longer than $L$L.
B
question 2
The number of babies born in a country each day is approximately normally distributed with mean $153$153 and standard deviation $28$28. The number of babies born on two consecutive days along with their $z$z-scores are provided below.
| Number of newborns | $265$265 | $69$69 |
|---|---|---|
| $z$z-scores | $4$4 | $-3$−3 |
Which event is more likely?
The number of babies born the following day is less than $69$69.
AThe number of babies born the following day is greater than $265$265.
B
question 3
The owner of a cafe records the arrival times of every customer each morning. The data set is approximately normally distributed with the busiest time at $9:00$9:00 am and a standard deviation of $14$14 minutes.
Complete the following table by finding the $z$z-scores of each arrival time.
Times $8:18$8:18 am $8:32$8:32 am $8:46$8:46 am $z$z-scores $\editable{}$ $\editable{}$ $\editable{}$ The cafe owner has a friend who can help for a short period of time in the morning. Which of the following is the busiest time period for her friend to help?
Before $8:18$8:18 am.
ABetween $8:18$8:18 am and $8:32$8:32 am.
BBetween $8:32$8:32 am and $8:46$8:46 am.
CBetween $8:46$8:46 am and $9:00$9:00 am.
D