Fill in the blanks in the statements below. The Central Limit Theorem states tha

Question

Fill in the blanks in the statements below. The Central Limit Theorem states that as the sample size increases, the distribution of all the possible sample means (that is the sampling distribution of sample means) approaches a ______________ distribution. The mean of the sampling distribution will be ______________ as/than the mean of the population and the standard deviation of the sampling distribution will be ______________ as/than the standard deviation of the population. (Fill in the first blank with one word. Fill each of the other blanks with a "comparison" word (or phrase). The walking gait of an adult male giraffe is normally distributed with a mean of 10 feet and a standard deviation of 1.5 feet. Complete the following. (a) Describe the shape and horizontal scaling on the graph of the distribution for the population of all adult male walking gaits (hereafter referred to simply as gaits). (b) Find the probability that the gait of a randomly selected adult male giraffe will be less than 8.5 feet---that is, find P(x < 8.5). Based upon your result, state whether or not it is unusual to randomly select an adult male giraffe whose gait is less than 8.5 (and explain why you chose "unusual" or "not unusual" as your answer). (c) Suppose all possible samples of size 36, taken from the population of all adult male giraffe gaits, are drawn and the mean is found for each resulting sample. Describe the shape and scaling on the graph of the resulting sampling distribution for the sample mean values. Hint: Apply the Central Limit Theorem! (d) Find the probability that the mean gait of a randomly selected sample of 36 adult male giraffes will be less than 9.2 feet---that is, find P(x-bar < 9.2). Based upon your result, state whether or not it is unusual to randomly select a sample of 36 adult male giraffes whose mean gait is less than 9.2 (and explain why you chose "unusual" or "not unusual" as your answer). (e) Find the probability that the mean from a sample of 36 gaits will be between 10.0 and 10.25 feet. Fill in the blanks in the statements below. The Central Limit Theorem states that as the sample size increases, the distribution of all the possible sample means (that is the sampling distribution of sample means) approaches a ______________ distribution. The mean of the sampling distribution will be ______________ as/than the mean of the population and the standard deviation of the sampling distribution will be ______________ as/than the standard deviation of the population. (Fill in the first blank with one word. Fill each of the other blanks with a "comparison" word (or phrase). The walking gait of an adult male giraffe is normally distributed with a mean of 10 feet and a standard deviation of 1.5 feet. Complete the following. (a) Describe the shape and horizontal scaling on the graph of the distribution for the population of all adult male walking gaits (hereafter referred to simply as gaits). (b) Find the probability that the gait of a randomly selected adult male giraffe will be less than 8.5 feet---that is, find P(x < 8.5). Based upon your result, state whether or not it is unusual to randomly select an adult male giraffe whose gait is less than 8.5 (and explain why you chose "unusual" or "not unusual" as your answer). (c) Suppose all possible samples of size 36, taken from the population of all adult male giraffe gaits, are drawn and the mean is found for each resulting sample. Describe the shape and scaling on the graph of the resulting sampling distribution for the sample mean values. Hint: Apply the Central Limit Theorem! (d) Find the probability that the mean gait of a randomly selected sample of 36 adult male giraffes will be less than 9.2 feet---that is, find P(x-bar < 9.2). Based upon your result, state whether or not it is unusual to randomly select a sample of 36 adult male giraffes whose mean gait is less than 9.2 (and explain why you chose "unusual" or "not unusual" as your answer). (e) Find the probability that the mean from a sample of 36 gaits will be between 10.0 and 10.25 feet.

Explanation / Answer

The Central Limit Theorem states that as the sample size increases, the distribution of all the possible sample means (that is the sampling distribution of sample means) approaches a ______Normal________ distribution. The mean of the sampling distribution will be _______same_______ as/than the mean of the population and the standard deviation of the sampling distribution will be _____less_________ as/than the standard deviation of the population.

The standard deviation will be sigma/root over n.

(b)For X=8.5, z=(x-mu)/sigma=(8.5-10)/1.5=-1

Thus the required probability is P(X<-1)=0.1587

The probability less than 0.05 is typically said to be unusual. Thus, the above probability is not unusual.

(d)The distribution of sample mean is also normal with mean mu=10 and standard deviation sigma/root over n=1.5/rot over 36=0.25.

Thus for x bar=9.2, z=(9.2-10)/0.25=-3.2

The required probability is P(x bar<9.2)=P(z<-3.2)=0.0007

The probability less than 0.05 is typically said to be unusual. Thus, the above probability is unusual.

(e)For X bar=10.25, z=(10.25-10)/0.25=1

For X bar=10, z=(10-10)/0.25=0

The required probability is:

P(10.25<X bar<10)=P(X bar<10.25)-P(X bar<10)

=P(z<1)-P(z<0)=0.8413-0.5=0.3413

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