DATA ANALYSIS: METEOROLOGY The table shows the monthly normal temperatures (in degrees Fahrenheit) for selected months in New York City $(N)$ and Fairbanks, Alaska $(F)$. (Source: National Climatic Data Center)
(a) Use the $regression$ feature of a graphing utility to find a model of the form $y = a\ sin(bt\ +\ c)\ +\ d$ for each city. Let $t$ represent the month, with $t=1$ corresponding to January.
(b) Use the models from part (a) to find the monthly normal temperatures for the two cities in February, March, May, June, August, September, and November.
(c) Compare the models for the two cities.
a) $N=22.099 \sin (0.522 t-2.219)+55.008$
$F=36.641 \sin (0.502 t-1.831)+25.610$
b) $N :$ Feb: $34.62,$ Mar: $41.58,$ May: $63.43,$ Jun: $72.50,$ Aug: $75.48,$ Sep: 68.60 Nov: 46.78
$F :$ Feb: $-1.35,$ Mar: $13.91,$ May: $48.62,$ Jun: $59.50,$ Aug: $55.55,$ Sep: 41.70 Nov: 6.48
c) The model for New York has a higher average temperature than the model for Fairbanks, and at any given time the temperature is higher in New York than it is in Fairbanks. The amplitude of the model for Fairbanks is larger than the amplitude of the model for New York, indicating a wider range of temperatures is found in Fairbanks than in New York.
Okay, we're using a graphing utility for this problem. So here's my graphing calculator, and we start by going into the stat menu for statistics and then into edit. And you can see I've already typed in the numbers. So for list one, these air the month numbers where January is month 1 April is month for July, as 7 October is 10 and December is 12 and then enlist to I have the temperatures for New York City, an endless three a Have the temperatures for Fairbanks. Okay, so the first thing we're going to do is find regression models or regression equations for each of these cities. So I press stat and then over to calculate. And then we're going to look for a sign regression model. And for this calculator, its way down the list. You just keep scrolling down past the more familiar regression models, and finally we get to sign regression. So I'm going to select that, and I want to use list one and list to if I'm using New York City and and I'm going to leave some of these things blank or unchanged. But what I do want to do is store my regression equation in my Y equals menu. So where it says store regression equation, I'm going to press variables and then over toe why? Variables and shoes function and choose why one and then I'm going to calculate. So what you see here are all the numbers that you would round and write down to have your model. We have the values of A, B, C and D. And then if we look in the y equals menu Ah, you should see the equation pasted in there. And I'm just going to erase that second equation that was left over from some other time. Okay, Now we want to find the regression equation for Fairbanks. So once again, I pressed at and then over to calculate, and then we're going to find sign regression again. And then this time I'm going to use list one and list three. So list one is for the month analysts threes for Fairbanks. So if I press second function three again l three and this one, I'm going to store as why, too. So I press variables over toe y variables, choose function, and then she was white, too. And then we calculate and these are all the values for that equation, so you would round them and write them down. And let's look at the y equals menu. And there we have it pasted in as well. Okay, so we have our models. So now we move on to Part B, and we're going to use these models to calculate the temperatures in different months. So I'm going to use the table for that. So first I'm going to go into my table set menu, which is second window. Make sure that it says independent ask, and that way you can type in whatever X values you want. Okay. From here, we're going into the table menu, which is second graf and we want February, which would be month to So we just type of to in the UAE. One column. We see the New York City temperature, and in the Y two column, we see the fair beings temperature and then march would be month three and in May would be month five and June would be six. Angela and August would be 8 September would be nine and November would be 11. So now we have all the months covered okay for Part C, it says, Compare the models, and I think it would be nice to look at the graphs in order to do that. And so let's go ahead and choose some window dimensions for these graphs. So press window. Remember, the X value represents the month, so we could go from zero to 13 and we would see a little bit more than a year. And then the Y value represents the temperature, so it gets pretty cold in Fairbanks. Let's go down to negative 20 and then it got up to 70 something in New York. So maybe we go up to 85 and perhaps we go by tens and let's take a look at that. So the first craft is showing us New York City's temperature over the course of a year, and then the next graph is going to show us Fairbanks. So some comparisons that we could make our that on any given day New York's temperature is going to be warmer than Fairbanks temperature, But we could also say that there's a greater spread vertically for Fairbanks. We call that the amplitude, and so there's more of a wide range of temperatures in Fairbanks, and then you could make other varied conclusions as well