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1. (45 marks) Short-Answer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answer. Unless otherwise stated, it is not necessary to simplify your answers in this question. a) Find the equation of the tangent plane to the graph of the function z = f(x,y) = sell at (0,1). Answer b) Evaluate (T, 0) if f(x,y) = sin(ry). Answer c) Graph of a function z = f(x,y) is given by: Which of the following level curves correspondes to this function. Answer d) If f(x,y) = e(3x – 2y), find lim - Math)-f(x,y) e) Find % if f(x, y) = x2 - y3 and 2 = r sin(t), y = r cos(t). Answer f) Evaluate the directional derivative of f(x, y) = !at (1, -1) in the direction of u =(2, 1). Answer 8) Does Simono converge? h) Sketch the direction field of the differential equation = 1-2y. i) Consider the differential equation ay = 15 - 31, y(0) = 0, and use Euler's method with step size 0.1 to estimate y(0.1). Answer j) Knowing that (-3,-3) is a critical point of f(x,y) = 2:3 + y2 – 4.cy -3.0, use the second derivative test to find out whether it is a local minimum, local maximum, or a saddle point. Answer k) Solve the initial value problem, dy dt 642 2y + cos(u) y(1) = . Answer 1) Is ū=(2,2, -1) perpendicular to v = (5,-4, 2)? Answer m) Find the area between the graph of y = 1 + sin(x) and y=1-sin(x) from 0 to 7. Answer n) Determine the lentgh of the curve y = 8/2,0 SIS 3. Evaluate and simplify your answer. Answer o) Express the volume of the solid obtained by rotating the bounded region that lies between curves y = (x-2)2 and y=1,150 $4 about the vertical line 2 = 5. Do not evaluate this integral. Answer

Standard Error of the Mean, Confidence Intervals, & The Z Test

Report the values you are using for this assignment.

HR Mean (µHR)

(one decimal)

HR Std Dev (sHR)

(two decimals)

RR Mean (µRR)

(one decimal)

RR Std Dev (sRR)

(two decimals)

90.7

6.83

12.6

1.96

Make sure these initial values have been marked correct.

Follow the rounding instructions carefully.

Task 1:  Standard Error of the Mean (SEM) for Different Sample Sizes

Imagine that your HR and RR data sets were very large (N=500). Calculate SEM based on the population standard deviations calculated in assignment 2, and the sample sizes given in the table. Carefully round the SEM value to 2 decimals for each calculation.

Formula: Standard Error of the Mean (SEM or sM or  ) = 

SEM

Heart Rate

Respiratory Rate

SEM if n = 9

2.28

0.65

SEM if n = 25

1.37

0.39

SEM if n= 100

0.68

0.2

What happens to the value of the SEM as the sample size increases?

By observing the values, we can conclude that as the sample size (n) increases the standard error of the mean decreases.

Question:

Task 5: Z Test: Calculate the Z statistic (Zobt) and compare to Zcrit

*Reminder! Your data is now viewed as a sample. Use the σ given in Task 4 to calculate SEM.

1) Use 3.16 as the square root of 10 to calculate the Standard Error of the Mean.  

(You may also see SEM expressed as  sM or  )

            (SEM or sM or  ) =       Round to 2 decimals.

2) Calculate the difference between the means (sample minus population)

!! If the difference is zero, use 1.1 as the difference.

3) Calculate ZOBT using:    

HR

SEM

ZOBT

Value

90.7 – 81.36

zHR = 2.08

RR

SEM

ZOBT

Value

12.6 – 15.69

zRR = -2.54

Alpha level (α) for all test is 0.05. The Zcrit values are:

Two Tailed

-1.96 and +1.96 (or ±1.96)

One Tailed

If difference between means is negative: -1.65

If difference between means is positive: +1.65

4) Compare ZOBT to ZCRIT. Is ZOBT located in the Zone of Rejection? (aka the Critical Region)

Sketching a number line with critical values and zones identified helps with this.

        Zone of                                                                                                           Zone of

       Rejection       -----------------------------Do Not Reject---------------------------------       Rejection

                      -1.96                                           0                                                 1.96                                                                                                                                                                                                                                                                        

Use the same principle for the one tailed comparison. In that case there will only be one Zone of Rejection, based on whether Zcrit was positive or negative.

Task 6: Z Test: Decisions, Effect Size, and Potential Error Type

1) Answer the first two questions (Yes or No) based on what you found in Task 2.

2) Calculate effect size using Cohen’s d:   d =      (Reported as a positive value.)

(Effect size is not usually reported if the null is not rejected, but do it here just to build skills.)

3) There is always a theoretical statistical error potentially present after a hypothesis is tested. Name the potential error that must be acknowledged because of your decision.

HR

Two Tail

One Tail

Is your decision to Reject the Null?

Can you say the sample mean is statistically different from the population mean?

What is the effect size? (Cohen’s d)

1.59

What type of statistical error might be present?

RR

Two Tail

One Tail

Is your decision to Reject the Null?

Can you say the sample mean is statistically different from the population mean?

What is the effect size? (Cohen’s d)

What type of statistical error might be present?

→ Consult the notes for detailed information on one and two tailed tests, hypothesis construction, alpha level, critical values, Zone of Rejection, types of error, and effect size by Cohen’s d.

→ The GWFW book uses the term Z ‘score’ even when comparing means; perhaps to highlight the similarity between the formula for a Z score (with sigma ‘σ’ in the denominator) and the formula for the Z statistic (with SEM in the denominator.) At the Z test level, a statistical test for comparing means is being performed by calculating a Z statistic. When the context is clear, ‘Z value’, or just ‘Z’, could be used as a general term of reference, but you should be aware that a Z score is not the same as a Z statistic.