More Algebra 2 problems.
June 2017, Part III
All Questions in Part III are worth up to 4 credits. Partial credit is possible.
Describe the behavior of the given function as x approaches -3 and as x approaches positive infinity.
35. Graph y = log2(x + 3) - 5 on the set of axes below. Use an appropriate scale to include
both intercepts.
Answer:
If you graph y = log2(x + 3) - 5, you will see that it is not defined for x %lt 3.
The y-intercept is at y = log2(0 + 3) - 5 = -3.42.
Solving 0 = log2(x + 3) - 5 gives you the x-intercept at 29.
You can use the calculator, or work it out:
5 = log2(x + 3)
x + 3 = 25
x + 3 = 32
x = 29
The best scale to use for the x-axis is 3. Use a scale of 1 for the y-axis.
Check the image below. The table of values is provided for you. It wasn't necessary for the exam, but it would be a good idea to label the intercepts.
Second part: As x approaches -3, the function goes to negative infinity. As x approaches infinity, the function approaches infinity.
Forty customers are selected randomly to undergo the new check-in procedure and the
proportion of customers who prefer the new procedure is 32.5%. The dealership decides not to
implement the new check-in procedure based on the results of the study. Use statistical evidence
to explain this decision.
36. Charlie’s Automotive Dealership is considering implementing a new check-in procedure for
customers who are bringing their vehicles for routine maintenance. The dealership will launch
the procedure if 50% or more of the customers give the new procedure a favorable rating when
compared to the current procedure. The dealership devises a simulation based on the minimal
requirement that 50% of the customers prefer the new procedure. Each dot on the graph below
represents the proportion of the customers who preferred the new check-in procedure, each of
sample size 40, simulated 100 times.
Assume the set of data is approximately normal and the dealership wants to be 95% confident of
its results. Determine an interval containing the plausible sample values for which the dealership
will launch the new procedure. Round your answer to the nearest hundredth.
Answer:
The mean is given as 0.506, and one standard deviation is 0.078. To be 95% confident in the result requires two standard deviations, or 2 * 0.078 above or below the mean. So the interval would be:
0.506 - 0.156 < 0.506 < 0.506 + 2 * 0.156
0.354 < 0.506 < 0.656
0.35 < 0.506 < 0.66
In the second part, since 32.5% is below 35.4%, the amount is outside of the 95% confidence interval.
Comments and questions welcome.
More Algebra 2 problems.
1 comment:
Thanks for all your work on this site.
In the question about Charlie's Automotive Dealership, I don't understand how the calculations leading to the official answer (which you explain clearly) satisfy the scenario. That is, I think the dealer is correct in not implementing the new procedure, but for an incorrect reason.
I interpret "dealership wants to be 95% confident of its results" as meaning that the dealership wants to have 95% confidence that "50% or more of the customers [...] prefer the new procedure". At least, that's what I would want if I were the dealer considering the change.
I'm not an expert in statistics, but it seems to me that:
The 95% confidence interval means that they have 95% confidence that, given that 50% of all customers prefer the new procedure, a random sample of 40 of them will have between 35% and 66% who prefer it. That's not the same thing, is it?
It looks like the dealership will implement the new procedure unless they have 95% confidence that the % of customers who prefer the new procedure is not 50%.
In the third part, when "Forty customers are selected randomly", what if, say, 36% of customers preferred the new procedure? Would they implement? Yes. Would they be 95% confident that 50% of customers prefer the new procedure? No way; more like 5% confidence.
I believe that to have that 95% confidence that 50% of customers will prefer the new procedure, around 65% of the 40 random customers in the sample would have to prefer it, right?
What's worse, what if, say, 67% preferred the new procedure? Would they not implement the new procedure because that's outside the 95% confidence interval?
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