Question```latex
\textbf{Problem:}
Given the following data on the average number of weekly hours worked by U.S. production workers from 1967 to 1996:
\begin{center}
\begin{tabular}{|c|c|}
\hline
Year & Hours Worked \\
\hline
1967 & 38.0 \\
1968 & 37.8 \\
1969 & 37.7 \\
1970 & 37.1 \\
1971 & 36.9 \\
1972 & 37.0 \\
1973 & 36.9 \\
1974 & 36.5 \\
1975 & 36.1 \\
1976 & 36.1 \\
1977 & 36.0 \\
1978 & 35.8 \\
1979 & 35.7 \\
1980 & 35.3 \\
1981 & 35.2 \\
1982 & 34.8 \\
1983 & 35.0 \\
1984 & 35.2 \\
1985 & 34.9 \\
1986 & 34.8 \\
1987 & 34.8 \\
1988 & 34.7 \\
1989 & 34.6 \\
1990 & 34.5 \\
1991 & 34.3 \\
1992 & 34.4 \\
1993 & 34.5 \\
1994 & 34.7 \\
1995 & 34.5 \\
1996 & 34.4 \\
\hline
\end{tabular}
\end{center}
1. Construct a scatter diagram and comment on the relationship, if any, between the variable Year and Hours Worked.
2. Determine and interpret the correlation for the year and hours worked. Based upon the value of the correlation, is your answer to the previous question reasonable?
3. Based upon the data given, estimate the average weekly hours worked this year. How confident are you in your estimate? You should use a linear regression model to make your prediction.
4. Assuming a linear correlation between these two variables, what will happen to the average weekly hours worked in the future? Is it possible for this pattern to continue indefinitely? Explain.
Studdy Solution
STEP 1
What is this asking?
We're looking at how the average weekly hours worked by U.S. production workers changed from 1967 to 1996, trying to see if there's a trend, and maybe even predict future work hours!
Watch out!
Just because two things seem related doesn't mean one *causes* the other.
Also, predictions way into the future can be tricky!
STEP 2
1. Scatter Diagram and Relationship
2. Correlation Calculation and Interpretation
3. Linear Regression and Prediction
4. Future Trend Analysis
STEP 3
Alright, let's **plot** these points!
We'll put the year on the horizontal -axis and the hours worked on the vertical -axis.
Each point represents a year and the corresponding average hours worked.
STEP 4
Looking at the scatter plot, it seems like there's a *downward trend*!
As the years go by, the average hours worked seem to be decreasing.
It looks pretty close to a straight line, so a linear relationship seems reasonable.
STEP 5
To **calculate** the correlation, we need the means and standard deviations of both the year and the hours worked.
Let's call the year and the hours worked .
STEP 6
After some number crunching, we find:
STEP 7
Now, we can **calculate** the correlation coefficient : Plugging in our values, we get .
STEP 8
Wow, that's a **strong negative correlation**!
This confirms what we saw in the scatter plot: as the years increase, the hours worked tend to decrease.
STEP 9
Let's **find** the equation of the regression line, which is of the form .
Here, and .
STEP 10
Using our calculated values, we get: So our regression equation is .
STEP 11
To **predict** the average hours worked for a given year, we just plug the year into our equation.
Let's try this year, say 2024:
STEP 12
So, our model predicts about **30.76 hours** worked per week this year.
However, we need to be careful with predictions too far outside our data range.
The world changes, and the trend might not continue forever!
STEP 13
If this linear trend were to continue, the average weekly hours worked would keep decreasing.
However, it's *unlikely* this pattern would continue indefinitely.
There's a limit to how low the average hours worked can go!
It can't go below zero.
Also, economic and social factors can influence working hours, and these factors can change over time.
STEP 14
The scatter diagram shows a strong negative linear relationship between year and average weekly hours worked.
The correlation coefficient of approximately confirms this strong negative relationship.
Based on the linear regression model, the estimated average weekly hours worked for 2024 is approximately 30.76 hours.
However, this prediction should be treated with caution as it extrapolates beyond the original data range.
It's unlikely the linear trend will continue indefinitely into the future.
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