The Ethan Allen tour boat capsized and sank in Lake George, New York, and 20 of the 47 passengers drowned. Based on a 1960 assumption of a mean weight of 140 lb for passengers, the boat was rated to carry 50 passengers. After the boat sank, New York State changed the assumed mean weight from 140 ib to 174 lb.
a. Given that the boat was rated for 50 passengers with an assumed mean of 140 tb, the boat had a passenger load limit of 7000 lb. Assume that the boat is loaded with 50 male passengers. and assume that weights of men are normally distributed with a mean of 189 lb and a standard deviation of 39 lb (based on Data Set 1 "Body Data" in Appendix B). Find the probability that the boat is overloaded because the 50 male passengers have a mean weight greater than 140 lb.
b. The boat was later rated to carry only 14 passengers, and the load limit was changed to 2436 lb. If 14 passengers are all males, find the probability that the boat is overloaded because their mean weight is greater than 174 lb (so that their total weight is greater than the maximum capacity of 2436 lb). Do the new ratings appear to be safe when the boat is loaded with 14 male passengers?
a. 0.8877 (Table 0.8869)$\\$
b. 1.0000 when rounded to four decimal places (Table: 0.9999) $\\$
The probability from part (a) is more relevant because it shows
that 89% of male passengers will not need to bend.$\\$
.Because men are generally taller than women, a design that
accommodates a suitable proportion of men will necessarily
accommodate a greater proportion of women
This problem is about about a boat accident where a boat sank in Lake George back in around 1960. And when that sank, they assumed that the average person wait on that boat. The assumed average was 140. Now, keep in mind when the passengers air on the boat, you're going to have male passengers. You're gonna have female passengers and you're going to have Children. Eso some of the people could have weighed more than 140 others could have weighed less. So in solving this problem, we're going to make an assumption that it was all males on the boat because they're usually ah, heavier population than the females or the Children. So throughout this problem, they gave us some information. They told us that men's weights are normally distributed. The average male weighs £189 and the standard deviation of the population of men would be £39. And there are two parts to this problem. The first part, let's look at part a in part, a. It's saying, given that the boat was rated for 50 passengers and they assumed that the mean weight was £140. Then the boat limit was £7000. So we're going to do a sample from the population of men and we're going to assume that there were 50 people on this boat and all of them were men. So we're talking about a sample size of 50 and we want to find the probability that the boat was overloaded. Well, if the boat is overloaded, that means the average passenger weighed more than 140. So in order to solve this, we're going to have to discuss the average of the sample means on the standard deviation of the sample means. And in order to calculate those we're going to use the central Limit theorem and the Central Limit Theorem says that the average of the sample means is going to be the equal to the average of the population, and in this scenario it was all men, so the average was £189. The standard deviation of the sample means will equal the standard deviation off the population divided by thes square root of the sample size. So in this case, it's going to be 39 divided by the square root of 50 because they told us that men's weights are normally distributed. We can draw our bell shaped curve, and we can put the average in the center at 189. And this problem is asking us to determine the probability that your average was greater than 140 now. In order to do this, we are going to have to use a C score, and we're going to have to use the Z score, associate it with sample means, and to refresh your memory, Z is going to equal X bar minus the average of the sample means over the standard deviation of the sample means so therefore, the Z score associated with 140 is going to be 140 minus 189. And in place of the standard deviation of the sample means we're going to use the expression 39 over the square root of 50 and we're going to calculate a C score out to be negative 8.88 I always like to put that up on the picture, so I'm gonna put negative 8.88 in correspondence with the 140. So when we're solving the problem where the probability is greater than 140 it's no different than saying, What's the probability that the Z score is greater than that negative 8.88 So you're going to need the standard normal table in the back of your textbook, which is table eight to. And if you look at the picture on that table, the picture shows that the area or the probability that we're trying to find extends into the left table or left tail and our picture is extending into the right tail. So we're going to have to rewrite our probability as one minus the probability that Z is less than negative 8.88 and again looking at that table, Um, when you're on the negative side of the table. If you wanted a new area or a probability, associate it with Z that is less than or equal to 3.50 it's going to be 0.0 001 So we're going to do one minus 10.1 and we're going to get a probability of 0.9999 So just summarizing part A. It was asking us what was the probability that the boat was overloaded. Being overloaded would mean that the average person wait more than £140 and that probability is 1400.9999 So after they discovered this, they changed the average weight. They made the assumption that you know what the average weight of the people on that boat was not 1 40 but they upped it to approximately 174 which takes us into part B in Part B. If they brought the average wait up to 174 and they downgraded the amount of passengers on the boat tow 14. So in part B, the boat was later rated to carry only 14 passengers. Um, they upped the estimate of the average person's weight, so it created a load limit of 2436. So now we're only gonna have 14 passengers were going to make the assumption there all male and again, we're using the mail because they're generally, um, heavier than females or the Children. And in Part B. We want to find the probability that those 14 passengers, how they mean wait greater than 174. So again, we are using a sample. This time, our sample size is only 14. We are going to need to calculate the average of the sample means and the standard deviation of the sample means again we're using our central limit theorem. The average of the sample means is going to equal the average of the population. Are population being the weights of men, which was 189. And our standard deviation of sample means according to the central limit, their, um will be equivalent to the standard deviation of the population divided by the square root of the sample size. Since our population is the weights of men are standard, deviation of men was £39. And this time we're going to divide by the square root of 14. Because our sample size is now only 14 passengers on the boat we're going to construct are bell shaped curve, gonna put our average in the center and we're trying to find the probability that were greater than on average weight of 1 74. We're going to use the same Z score formula. So we're going to use the Z score formula Z equals X bar minus mu sub X bar over Sigma sub X bar. So we want the Z score, associate it with 174. So we're gonna do C equals 1 74 minus 189 in place of the standard deviation of the sample means we're going to use the expression 39 divided by the square root of 14. And in doing so are Z score turns out to be negative 1.44 again, I like to always put that z score back on the picture. So we're gonna put negative 1.44 here. Keep in mind in part B, we were trying to find the probability that the average was greater than £174. So when we're going greater than £174 it's no different than saying What's the probability that Z is greater than negative 1.44 again, our tables were set up to go into the left tail. Our picture is going into the right tail, so we're going to have to rewrite our probability as one, minus the probability that sees less than negative 1.44 We're going to go to our standard normal table, which is tabled a to, and we're going to get the probability that see us less than negative 1.44 to be 0.749 And when we subtract that, we get an overall probability of 0.9 to 51 So in summary of part B, we readjusted theme passenger limit down to 14. We made a new assumption that the average passenger weighed 174 and we wanted to, um, know what the probability that the average weight of the 14 passengers on the boat was greater than 174 and that was 1740.9 to 51 Now, in part B, there is a second question, and that question says, Do the new ratings appear to be safe when the boat is loaded with 14 male passengers? And our answer to that would be because there's still a high probability, and that probability is 0.9 to 51 That's pretty high. The new ratings do not appear safe when the boat is loaded with 14 male passengers