2. A researcher measures the heart rate in a population of runners and finds they have an average heart rate of µ = 50, σ = 5. You do not need to show a graph, only computation work.
2A. Based on the average heart rate cited above, what proportion would we expect to have a heart rate of 55 or more? Show your Work! (2 points)
2B. Based on the average heart rate cited above, what proportion would we expect to have a heart rate of 40 or less? Show your Work! (2 points)
2C. Based on the average heart rate cited above, what proportion would we expect to have a heart rate between 55 and 65? Show your Work! (3 points)
2D. Based on the average heart rate cited above, what heart rate would we expect to cut off the top 10% of the distribution? Show your Work! (2 points)
3. A group of toys consists of the following:
5 dolls
7 stuffed animals
3 toy guns
3A. If I choose at random from the group of toys, what is the probability I get a doll--P(doll)?
Show your Work! (2 points)
1/3
3B. If I choose at random from the group of toys, what is the probability I get a doll or a gun--P(doll or gun)? Show your Work! (2 points)
3C. If I choose two toys at random (with replacement), what is the probability I get a doll and a stuffed animal—P (doll and stuffed animal)? Show your Work! (2 points)
3D. If take out toys and give them away as I take them (without replacement), what is the probability I get a doll, a stuffed animal, and a gun—P(doll and stuffed animal and gun)?
Show your Work! (2 points)
for 2 use :
http://davidmlane.com/hyperstat/z_table.html
3 a
Yes, 5/15 = 1/3
3 b
just add 5/15 + 3/15 = 8/15
3 c
p
what are chances of
d then a then g ?
5/15 * 7/14 * 3/13 = .03846153
d then g then a ?
5/15 * 3/14 * 7/13 = same
etc
a then d then g
a then g then d
g then d then a
g then a then d
six ways in total, each .03846153
so multiply by 6 and get
.231
2A. To find the proportion of runners with a heart rate of 55 or more, we need to find the area under the normal distribution curve to the right of 55. We can use the z-score formula to calculate this.
First, we calculate the z-score:
z = (55 - µ) / σ
z = (55 - 50) / 5
z = 1
Next, we look up the z-score in the z-table to find the proportion. From the z-table, we can see that the proportion corresponding to a z-score of 1 is approximately 0.8413.
Therefore, the proportion we would expect to have a heart rate of 55 or more is approximately 0.8413 or 84.13%.
2B. To find the proportion of runners with a heart rate of 40 or less, we need to find the area under the normal distribution curve to the left of 40.
First, we calculate the z-score:
z = (40 - µ) / σ
z = (40 - 50) / 5
z = -2
Next, we look up the z-score in the z-table to find the proportion. From the z-table, we can see that the proportion corresponding to a z-score of -2 is approximately 0.0228.
Therefore, the proportion we would expect to have a heart rate of 40 or less is approximately 0.0228 or 2.28%.
2C. To find the proportion of runners with a heart rate between 55 and 65, we need to find the area under the normal distribution curve between these two values.
First, we calculate the z-scores for the lower and upper bounds:
z1 = (55 - µ) / σ
z1 = (55 - 50) / 5
z1 = 1
z2 = (65 - µ) / σ
z2 = (65 - 50) / 5
z2 = 3
Next, we look up the z-scores in the z-table to find the proportions. From the z-table, we can see that the proportion corresponding to a z-score of 1 is approximately 0.8413, and the proportion corresponding to a z-score of 3 is approximately 0.9987.
We then subtract the smaller proportion from the larger proportion to find the area between the two z-scores:
Proportion = 0.9987 - 0.8413 = 0.1574
Therefore, the proportion we would expect to have a heart rate between 55 and 65 is approximately 0.1574 or 15.74%.
2D. To find the heart rate that would cut off the top 10% of the distribution, we need to find the z-score that corresponds to a cumulative proportion of 0.9. This z-score will give us the desired heart rate.
From the z-table, we can see that the z-score corresponding to a cumulative proportion of 0.9 is approximately 1.28.
Next, we use the z-score formula to find the heart rate:
z = (x - µ) / σ
1.28 = (x - 50) / 5
6.4 = x - 50
x = 56.4
Therefore, the heart rate we would expect to cut off the top 10% of the distribution is approximately 56.4.