Hello Jamie! For whatever reason, I am still up and decided to answer your Statistics question regarding normal density curve. My assumption is that you have in your possession a graphing calculator such as the TI-84, which is what I'm using for my calculations. If you don't have a TI-84, you'd have to use tables to find many of these answers....ugghhh!
Well, let's start at the top. The question states a uniform distribution, which is another way of saying Normal distribution, which means that you can utilize the Empirical Rule or number line to work backwards to find your mean and standard deviation. By finding the midpoint of 14 and 35, you'd get the mean (14 + 35)/2 or 24.5. I used the guess and check method to find the standard deviation of 3.5. You can check this by starting at -3 Standard deviations (14) and adding 3.5 until you reach the end of your number line at +3 Standard deviations(35).
The two probability questions can be answered by using Normalcdf in the calculator. My parameters for the P (16<X<22) were Normalcdf (16, 22, 24.5, 3.5), which provided an area of .2299 or 22.99%. I'll let you calculate the other one by using the same method. The two percentile questions can be calculated by using the InvNorm in the calculator. For the 60th percentile my parameters are InvNorm (.60, 24.5, 3.5), which provides an X value of 25.387. Since we are talking about cars, it'd make sense to round up to 26 cars because it makes no sense having a fractional amount of cars and rounding down would be less than the 60th percentile. I'll let you calculate the 65th percentile by using the same method. I'll give you a hint by saying the answer is 25.xxxx. You'll have to fill in the blanks.
Now that last question is pretty interesting because conditional probability is being inserted into this section for Normal Distributions, which is uncommon. However, the answer can be calculated by finding the probability of the condition, which is your denominator, and the intersection of the two events would be your numerator. Finding your denominator would be Normalcdf (19, 9999, 24.5, 3.5) or .9420 and your numerator would be Normalcdf (27, 9999, 24.5, 3.5) or .2375. The conditional fraction would be (.2375/.9420) or a final answer of .2521 or 25.21%. To make sure you understand what this answer is saying, I 'll put it in context for you. Speeds more than 27mph represents 25.21% of all speeds that are at least 19 mph. Well, Jamie I hope that helps. I'm going to bed now and I see that you are in Augusta, Georgia and I'm a double UGA alum(BBA '95,MBed '98), so I'll end this post by saying GO DAWGS!!!
