Question

The amount of time that you have to wait before seeing the doctor in the doctor's office is normally distributed with a mean of 15.2 minutes and a standard deviation of 15.2 minutes. If you take a random sample of 64 patients, what is the probability that the average wait time is greater than 20 minutes?

Answer 1

The amount of waiting before seeing a doctor is normally distributed with mean = 15.2 minutes and standard deviation = 15.2 minutes

Since we have to find probabilty of average wait time greater than 20 minutes, we will find z value for this sample mean and then corresponidng p value, since this p value will show probability of seeing doctor in less than or equal to 20 minutes, so we will substract this value from 1 and then our required value will appear.

So sample size = n = 64

Sample mean = X = 20 minutes

Population mean = μ = 15.2 minutes

Population standard deviation = σ = 15.2 minutes

$z= \frac{X-\mu}{\frac{\sigma}{\sqrt{n}}}$

$z= \frac{20-15.2}{\frac{15.2}{\sqrt{64}}}$

p value for z = 2.526 is 0.9942, which means probabilty for average waiting time less than or equal to 20 minute is 0.9942

So for probabilty of average waiting time to be greater than 20 minute is

=1-0.9942 = 0.0058

I have also shown in z table , corresponding p value,

Number in the table represents PIZsz) 02 Z 0.01 0.04 0.05 0.06 0.07 0.08 0.09 0.00 .5000 .5040 .5160 .5199 5319 .5359 0.0 0.1 .5239 5636 .5438 .5557 .5596 .5714 .5753 .5398 .5793 .6179 0.2 5279 .5675 .6064 .6443 0.02 .5080 .5478 .5871 .6255 .6628 .6985 .5832 .5948 .6026 .6141 .5987 .6368 .6103 .6480 0.3 .6217 .6517 0.4 .6554 .6591 .6331 .6700 .7054 .6736 .6808 .6844 .6879 .6915 .6950 .7088 .7190 .7224 0.5 0.6 7257 .7291 .7389 .7422 .7517 .7549 .7157 .7486 .7794 .8078 0.7 ,7704 .7734 .7823 .7611 .7910 .7324 .7642 .7939 .8212 .7852 .8133 0.8 .7995 .8023 .7580 .7881 .8159 .8413 .8106 .8186 .8264 .8289 .8340 .8365 .8389 0.9 1.0 1.1 .8438 .8461 .8508 .8531 .8577 .8621 .8599 .8810 .8686 .8749 .8790 .8830 .8643 .8849 .9032 .8729 .8925 1.2 .8888 .8944 .8980 .8997 .9015 0.03 .50 .5.17 .5. 10 .6193 .654 .719 .7 57 .773 .757 .8 38 .8 85 .8 p8 .87 .932 .9136 .970 .9 84 .982 .954 .932 .988 .934 .971 .901 .925 -9943 .9957 1.3 .8665 .8869 .9049 .9207 .9345 .9463 .9066 .9147 .9292 1.4 .9192 .9222 .9357 1.5 .6406 .6772 .7123 .7454 .7764 .8051 .8315 .8554 .8770 .8962 .9131 .9279 .9406 .9515 .9608 .9686 .9750 .9803 .9846 .9881 .9909 .9931 .9948 .9961 .9099 .9251 .9382 .9495 .9591 .9671 .9332 .9115 .9265 .9394 .9505 .9599 1.6 .9452 .9474 1.7 .9177 .9319 .9441 .9545 .9633 .9706 .9767 .9817 .9554 .9564 .9418 .9525 .9616 .9693 .9756 .9808 .9573 .9162 .9306 .9429 .9535 .9625 .9699 .9761 .9812 .9854 1.8 .9649 .9656 .9678 1.9 .9738 .9793 .9744 .9798 .9842 .9857 2.0 2.1 2.2 2.3 .9641 .9713 .9772 .9821 .9861 .9893 .9918 0000 .9953 .9719 .9778 .9826 .9864 .9896 .9920 .9726 .9783 .9830 .9868 .9898 .9922 .9878 .9887 .9890 .9916 .9936 .9850 .9884 .9911 .9932 .9949 .9962 2.4 .9838 .9875 .9904 .9927 .9945 .9959 .9969 .9977 .9984 2.5 .9913 .9934 .9951 .9963 0010 .9955 .9966 .9975 2.6 2.7 .9906 .9929 .9946 .9960 .9970 .9978 .9984 .9965 .9968 .9972 .9973 .9956 .9967 .9976 .9982 .9987 .9971 .9979 .9952 .9964 .9974 .9981 .9986 .9990 2.8 .9974 .9979 .9980 2.9 .9981 .9982 .9985 .9985 9986 3.0 .9987 .9988 .9990 3.1 .9987 .9990 .9993 .9991 .9992 .9991 .9993 .9993 .9989 .9992 .9994 3.2 .9977 .9983 .9988 .9991 .9994 .9996 .9997 .9998 .9999 .9994 .9989 .9992 .9994 .9996 .9997 .9998 .9993 .9995 .9996 .9995 .9997 3.3 .9995 .9996 .9995 .9997 .9989 .9992 .9995 .9996 .9997 .9998 .9999 3.4 .9994 .9996 .9997 .9998 .9999 .9995 .9997 .9998 .9998 9997 .9998 3.5 .9997 .9998 .9999 .9998 .9997 .9998 .9999 .9998 .9998 .9999 3.6 .9998 9999 .9999