calculate the least-squares line of best fit---**Table Content:**
| Column Header | Data 1 | Data 2 | Data 3 | Data 4 | Data 5 | Data 6 | Data 7 | Data 8 |
|---|---|---|---|---|---|---|---|---|
| Number of Tourists (000's) | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 |
| Temperature (in°Celsius) | 6 | 8 | 11 | 15 | 18 | 20 | 21 | 25 |
**Question(s):**
a. Find Pearson's correlation coefficient to three decimal places. R= 0.993
b. Comment on the association between temperature and number of tourists.
c. Calculate the least-squares line of best fit.
d. Construct a scatterplot for the table with temperature as the horizontal axis.
e. Predict the number of tourists in the country if the average maximum temperature is 0°C.
a. Predict the number of tourists in the country if the average maximum is 35°C.
f. Do you think this prediction is accurate? Why or why not?
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Today we'll calculate the least-squares line of best fit for data showing the relationship between temperature and number of tourists. We have 8 data points where temperature ranges from 6 to 25 degrees Celsius, and tourist numbers range from 10 to 24 thousands. Let's start by plotting these points on a scatter plot.
To calculate the least-squares line, we first need to compute several sums from our data. For each data point, we calculate x times y and x squared. The sum of all x values is 124, sum of y values is 136, sum of xy products is 2336, and sum of x squared is 2236. We have 8 data points total. These sums will be used in the formulas for slope and y-intercept.
Now we calculate the slope using the formula m equals n times sum xy minus sum x times sum y, all divided by n times sum x squared minus sum x squared. Substituting our values: 8 times 2336 minus 124 times 136, divided by 8 times 2236 minus 124 squared. This gives us 18688 minus 16864 over 17888 minus 15376, which equals 1824 over 2512, resulting in a slope of 0.7261. For the y-intercept, we use b equals sum y minus m times sum x, all divided by n. This gives us 136 minus 0.7261 times 124, divided by 8, which equals 5.7455. Therefore, our least-squares line is y equals 0.7261x plus 5.7455.
Here we see the scatterplot with our calculated line of best fit. The red line represents the equation y equals 0.7261x plus 5.7455. Notice how the line passes close to most of the data points, showing a strong positive linear relationship between temperature and the number of tourists. As temperature increases, the number of tourists tends to increase as well. The line provides a good fit to the data, which we can use to make predictions.
Now let's make predictions using our equation. For 0 degrees Celsius, we substitute x equals 0 into our equation: y equals 0.7261 times 0 plus 5.7455, which gives us approximately 5,746 tourists. For 35 degrees Celsius, we get y equals 0.7261 times 35 plus 5.7455, which equals 31,159 tourists. However, we must consider prediction accuracy. The 0 degree prediction may be reasonable since it's close to our data range. The 35 degree prediction is less reliable because it's far outside our observed temperature range of 6 to 25 degrees. Extrapolation beyond the data range can lead to unrealistic predictions. Our final least-squares line of best fit is y equals 0.7261x plus 5.7455.