The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 37 o
unces and a standard deviation of 7 ounces. Use the Empirical Rule, also known as the 68-95-99.7 Rule. Do not use Tables or Technology to avoid rounding errors. Suggestion: sketch the distribution in order to answer these questions. a) 68% of the widget weights lie between and b) What percentage of the widget weights lie between 23 and 44 ounces
1 answer:
Answer:
a) 68 % will lie between 30 and 44 ounces
b) between 23 and 44 ounces will lie 81,5 %
Step-by-step explanation: See Annex
a) Normal Distribution N ( 37, 7)
The Empirical Rule establishes that 68 % of values will be at
μ ± 1 σ where μ is the mean and σ the standard deviation
Then: 37 - 7 = 30 and 37 + 7 = 44
68 % of the values will lie between 30 and 44 ounces
b) And between 23 and 44 we should find:
We have 95 % of values between
μ ± 2*σ
In our case to the left of the value (mean 0 ) and up to the value 23 we get 95/2 = 47,5 % and to the right of the value (mean 0 ) and up to 44 we half 68/2 = 34 %
Then between 23 and 44, we have 47,5 + 34 = 81,5 %
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Part A:
x + y = 9
4x + y = 6
To solve this system using substitution, begin by isolating either x or y in the first equation. I'll isolate x.
x + y = 9
x = 9 - y
Substitute this expression (9 - y) into the second equation for x and solve.
4x + y = 6
4(9 - y) + y = 6
36 - 4y + y = 6
36 - 3y = 6
-3y = -30
y = 10
To find x, substitute 10 for y into either of the original equations.
x + y = 9
x + 10 = 9
x = -1
Finally, check all work by substituting the x- and y-values into each original equation.
x + y = 9 --> -1 + 10 = 9 --> 9 = 9 --> True
4x + y = 6 --> 4(-1) + 10 = 6 --> -4 + 10 = 6 --> 6 = 6 --> True
The answer for Part A is x = -1 and y = 10; (-1, 10).
Part B:
For graphing, it's easier to get the equations into slope-intercept form. Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. To get our equations into slope-intercept form, we must simply isolate y.
x + y = 9
y = 9 - x
y = -x + 9
4x + y = 6
y = 6 - 4x
y = -4x + 6
Now that we have our slope-intercept equations, we can easily graph them, since their slopes and y-intercepts are readily visible.
Let's start with the y-intercepts. They are (0, 9) and (0, 6). You can plot those points on the graph.
Now, the slopes. The slope of the first line is -1, this means it declines. To plot this, start where you plotted the y-intercept, count one unit down, and then one unit to the right, and plot that point. Continue doing that and connect the dots, and you will have plotted the first line. The slope of the second line is -4, so it also declines. For this line, count four units down, and then one to the right and plot that point. Likewise, continue this and connect the coordinates, and you will have your line. (See attachment for graph.)
The lines do indeed intersect at (-1, 10); our answer is verified by graphing.
