Let ŷ represent the profit (or loss) for a certain company x years after 1960. Based on the data shown below, a statistician calculates a linear model ŷ = – 2.31x + 49.22. 42.9 4 40.6 38.3 6. 33.9 7 32 8 31.3 9. 27.3 10 26.1 11 24.7 12 23.4 13 18.3 14 16.8 15 13.8 16 13.3 17 9.1 18 8.4 Use the model to estimate the profit in 1963. Round to the nearest cent. The estimated profit in 1963 is $
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
![### Linear Model for Company Profit Estimation
Let \(\hat{y}\) represent the profit (or loss) for a certain company \(x\) years after 1960. Based on the data shown below, a statistician calculates a linear model \(\hat{y} = -2.31x + 49.22\).
#### Data Table
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
3 & 42.9 \\
4 & 40.6 \\
5 & 38.3 \\
6 & 33.9 \\
7 & 32.0 \\
8 & 31.3 \\
9 & 27.3 \\
10 & 26.1 \\
11 & 24.7 \\
12 & 23.4 \\
13 & 18.3 \\
14 & 16.8 \\
15 & 13.8 \\
16 & 13.3 \\
17 & 9.1 \\
18 & 8.4 \\
\hline
\end{array}
\]
#### Linear Model
The linear model calculated is given by the equation:
\[
\hat{y} = -2.31x + 49.22
\]
### Estimation Task
**Problem:** Use the model to estimate the profit for the year 1963. Round to the nearest cent.
**Calculation:**
The number of years after 1960 for the year 1963 is \( x = 1963 - 1960 = 3 \).
Using the model \(\hat{y} = -2.31x + 49.22\):
\[
\hat{y} = -2.31(3) + 49.22
\]
\[
\hat{y} = -6.93 + 49.22
\]
\[
\hat{y} = 42.29
\]
The estimated profit in 1963 is \(\$42.29\).
**Answer:** The estimated profit in 1963 is $42.29.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcebbde0b-56df-456f-b45f-057baf1d3a0f%2F72413be3-139f-4c79-9ca1-9ea98f2af077%2Fubcuabh_processed.png&w=3840&q=75)
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