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Why is probability theory important to statistics?

In Exercises 9鈥�12, refer to the exercise identified. Make subjective estimates to decide whether results are significantly low or significantly high, then state a conclusion about the original claim. For example, if the claim is that a coin favours heads and sample results consist of 11 heads in 20 flips, conclude that there is not sufficient evidence to support the claim that the coin favours heads (because it is easy to get 11 heads in 20 flips by chance with a fair coin).

Exercise 8 鈥淧ulse Rates鈥�

Vitamin C and Aspirin A bottle contains a label stating that it contains Spring Valley pills with 500 mg of vitamin C, and another bottle contains a label stating that it contains Bayer pills with 325 mg of aspirin. When testing claims about the mean contents of the pills, which would have more serious implications: rejection of the Spring Valley vitamin C claim or rejection of the Bayer aspirin claim? Is it wise to use the same significance level for hypothesis tests about the mean amount of vitamin C and the mean amount of aspirin?

Fill in the blanks.

(a) A is a quantitative variable whose value depends on chance.

(b) A discrete random variable is a random variable whose possible values .

If you sum the probabilities of the possible values of a discrete random variable, the result always equals .

In 10 Bernoulli trials, how many outcomes contain exactly three successes?

Craps. The game of craps is played by rolling two balanced dice. A first roll of a sum of 7 or 11 wins; and a first roll of a sum of 2,3 , or 12 loses. To win with any other first sum, that sum must be repeated before a sum of 7 is thrown. It can be shown that the probability is 0.493 that a player wins a game of craps. Suppose we consider a win by a player to be a success, $s$

a. Identify the success probability, $p$

b. Construct a table showing the possible win-lose results and their probabilities for three games of craps. Round each probability to three decimal places.

c. Draw a tree diagram for part (b).

d. List the outcomes in which the player wins exactly two out of three times.

e. Determine the probability of each of the outcomes in part (d). Explain why those probabilities are equal.

f. Find the probability that the player wins exactly two out of three times.

g. Without using the binomial probability formula, obtain the probability distribution of the random variable $Y$, the number of times out of three that the player wins.

h. Identify the probability distribution in part (g).

Expressing Confidence Intervals Example 2 showed how the statistics of n= 22 ands= 14.3 result in this 95% confidence interval estimate of \(\sigma \): 11.0 < \(\sigma \) < 20.4. That confidence interval can also be expressed as (11.0, 20.4), but it cannot be expressed as 15.7 \( \pm \) 4.7. Given that 15.7\( \pm \)4.7 results in values of 11.0 and 20.4, why is it wrong to express the confidence interval as 15.7\( \pm \)4.7?

Normality Requirement What is different about the normality requirement for a confidenceinterval estimate of \(\sigma \)and the normality requirement for a confidence interval estimateof \(\mu \)?

ASU Enrollment Summary. According to the Arizona State University Enrollment Summary, a frequency distribution for the number of undergraduate students attending Arizona State University ( ASU ) in the Fall 2012 semester, by class level, is as shown in the following table. Here , 1 = freshman , 2 = sophomore , 3 = junior , and 4 = senior.

Let X denote the class level of a randomly selected ASU undergraduate.

(a) What are the possible values of the random variable X?

(b) Use random-variable notation to represent the event that the student selected is a junior ( class - level 3 ).

(c) Determine P ( X = 3 ), and interpret your answer in terms of percentages.

(d) Determine the probability distribution of the random variable X.

(e) Construct a probability histogram for the random variable X.