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Statistics For Business And Economics

James T. McClave, P. George Benson, Terry Sincich

$Math Studyset Vaia Explanations$ Math

13th Edition · 888 pages

ISBN: 9780134506593

Chapter 8: Q8E (page 452)

What is the line of means?

Short Answer

Expert verified

The equation y= β₀+ β₁x is referred to as the line of means in the Probabilistic model.

Step by step solution

01

Introduction.

A probabilistic model has both a deterministic and a random error component. For example, suppose we anticipate that sales y are related to advertising spend x by

y = 15x + Random error

General Form of Probabilistic Models:

y = Deterministic component + Ramdom error,

Where, y is the variable of interest we always assume that the mean value of the random mistake is 0. This is comparable to assuming that the mean value of y, E(y), equals the model's deterministic component; that is,

E(y) = Deterministic component.

02

Brief explanation to the line of means.

We know that a basic linear regression equation has the form, where y= β₀+ β₁x is the dependent variable,x is the independent variable, β₀is the intercept, β₁is the slope ε and is an error random variable with mean 0 and variance of . The probabilistic model will then bey= β₀+ β₁x. If we consider E(y), we may determine that

E(y)= β₀+ β₁x.

Every y value is normally distributed with mean values based on x for each value of x. When we plot this E(y) with x, we obtain a straight line with all mean lying on it; this line is known as the line means.

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Most popular questions from this chapter

Intrusion detection systems. The Journal of Researchof the National Institute of Standards and Technology (November–December 2003) published a study of a doubleintrusion detection system with independent systems. Ifthere is an intruder, system A sounds an alarm with probability.9, and system B sounds an alarm with probability.95. If there is no intruder, system A sounds an alarm withprobability .2, and system B sounds an alarm with probability.1. Now assume that the probability of an intruderis .4. Also assume that under a given condition (intruderor not), systems A and B operate independently. If bothsystems sound an alarm, what is the probability that anintruder is detected?

Question: Refer to the Bulletin of Marine Science (April 2010) study of lobster trap placement, Exercise 6.29 (p. 348). Recall that the variable of interest was the average distance separating traps—called trap-spacing—deployed by teams of fishermen. The trap-spacing measurements (in meters) for a sample of seven teams from the Bahia Tortugas (BT) fishing cooperative are repeated in the table. In addition, trap-spacing measurements for eight teams from the Punta Abreojos (PA) fishing cooperative are listed. For this problem, we are interested in comparing the mean trap-spacing measurements of the two fishing cooperatives.

BT Cooperative	93	99	105	94	82	70	86
PA Cooperative	118	94	106	72	90	66	98

Source: Based on G. G. Chester, “Explaining Catch Variation Among Baja California Lobster Fishers Through Spatial Analysis of Trap-Placement Decisions,” Bulletin of Marine Science, Vol. 86, No. 2, April 2010 (Table 1).

a. Identify the target parameter for this study.b. Compute a point estimate of the target parameter.c. What is the problem with using the normal (z) statistic to find a confidence interval for the target parameter?d. Find aconfidence interval for the target parameter.e. Use the interval, part d, to make a statement about the difference in mean trap-spacing measurements of the two fishing cooperatives.f. What conditions must be satisfied for the inference, part e, to be valid?

Question: Two independent random samples have been selected—100 observations from population 1 and 100 from population 2. Sample means ${\bar{x}}_{1} =26.6, {\bar{x}}_{2} = 15.5$ were obtained. From previous experience with these populations, it is known that the variances are ${σ_{1}}^{2} = 9 a n d {σ_{2}}^{2} = 16$ .

a. Find $σ_{({\bar{x}}_{1} - {\bar{x}}_{2})}$ .

b. Sketch the approximate sampling distribution for $({\bar{x}}_{1} - {\bar{x}}_{2})$ , assuming $(μ_{1} - μ_{2}) = 10$ .

c. Locate the observed value of $({\bar{x}}_{1} - {\bar{x}}_{2})$ the graph you drew in part

b. Does it appear that this value contradicts the null hypothesis $H_{0} : (μ_{1} - μ_{2}) = 10$ ?

d. Use the z-table to determine the rejection region for the test against $H_{0} : (μ_{1} - μ_{2}) \neq 10$ . Use $α = 0.5$ .

e. Conduct the hypothesis test of part d and interpret your result.

f. Construct a confidence interval for $(μ_{1} - μ_{2})$ . Interpret the interval.

g. Which inference provides more information about the value of $(μ_{1} - μ_{2})$ — the test of hypothesis in part e or the confidence interval in part f?

A random sample of n = 6 observations from a normal distribution resulted in the data shown in the table. Compute a 95% confidence interval for $σ^{2}$

Optimal goal target in soccer. When attempting to score a goal in soccer, where should you aim your shot? Should you aim for a goalpost (as some soccer coaches teach), the middle of the goal, or some other target? To answer these questions, Chance (Fall 2009) utilized the normal probability distribution. Suppose the accuracy x of a professional soccer player’s shots follows a normal distribution with a mean of 0 feet and a standard deviation of 3 feet. (For example, if the player hits his target,x=0; if he misses his target 2 feet to the right, x=2; and if he misses 1 foot to the left,x=-1.) Now, a regulation soccer goal is 24 feet wide. Assume that a goalkeeper will stop (save) all shots within 9 feet of where he is standing; all other shots on goal will score. Consider a goalkeeper who stands in the middle of the goal.

a. If the player aims for the right goalpost, what is the probability that he will score?

b. If the player aims for the center of the goal, what is the probability that he will score?

c. If the player aims for halfway between the right goal post and the outer limit of the goalkeeper’s reach, what is the probability that he will score?

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