MATHEMATICS COLLEGE

Suppose a simple random sample of size nequals64 is obtained from a population with mu equals 88 and sigma equals 8. (a) Describe the sampling distribution of x overbar. (b) What is Upper P (x overbar greater than 89.7 )? (c) What is Upper P (x overbar less than or equals 85.7 )? (d) What is Upper P (87.35 less than x overbar less than 90.5 )?

Answers

Answer 1

Answer:

Answer:

a) $\bar X \sim N (\mu, (\sigma)/(√(n)))$

With:

$\mu_(\bar X)= 88$

$\sigma_(\bar X)= 8$

b) $z=(89.7-88)/((8)/(√(64)))= 1.7$

$P(Z>1.7) = 1-P(Z<1.7) =1-0.955=0.0446$

c) $z =(85.7-88)/((8)/(√(64)))= -2.3$

$P(Z<-2.3) = 0.0107$

d) $z =(87.35-88)/((8)/(√(64)))= -0.65$

$z =(90.5-88)/((8)/(√(64)))= 2.5$

$P(-0.65<z<2.5)=P(Z<2.5)-P(Z<-0.65) =0.994-0.258 = 0.736$

Step-by-step explanation:

For this case we know the following propoertis for the random variable X

$\mu = 88, \sigma = 8$

We select a sample size of n = 64

Part a

Since the sample size is large enough we can use the central limit distribution and the distribution for the sample mean on this case would be:

$\bar X \sim N (\mu, (\sigma)/(√(n)))$

With:

$\mu_(\bar X)= 88$

$\sigma_(\bar X)= 8$

Part b

We want this probability:

$P(\bar X>89.7)$

We can use the z score formula given by:

$z = (\bar X -\mu)/((\sigma)/(√(n)))$

And if we find the z score for 89.7 we got:

$z=(89.7-88)/((8)/(√(64)))= 1.7$

$P(Z>1.7) = 1-P(Z<1.7) =1-0.955=0.0446$

Part c

$P(\bar X<85.7)$

We can use the z score formula given by:

$z = (\bar X -\mu)/((\sigma)/(√(n)))$

And if we find the z score for 85.7 we got:

$z =(85.7-88)/((8)/(√(64)))= -2.3$

$P(Z<-2.3) = 0.0107$

Part d

We want this probability:

$P(87.35 <\bar X< 90.5)$

We find the z scores:

$z =(87.35-88)/((8)/(√(64)))= -0.65$

$z =(90.5-88)/((8)/(√(64)))= 2.5$

$P(-0.65<z<2.5)=P(Z<2.5)-P(Z<-0.65) =0.994-0.258 = 0.736$

Related Questions

Pls answer asapA traveler has 9 pieces of luggage. How many ways can the traveler select 3 pieces of luggage for a trip? Responses 84 84 504 504 60,480 60,480 362,880
Help pls is my homework helppppppppp 15 points
An element with a mass of 570 grams decays by 26.7% per minute. To the nearest minute, how long will it be until there are 40 grams of the element remaining?
If m = 1/2, n= 1/4 , then m²+ n²
In a study of factors affecting whether soldiers decide to reenlist, 320 subjects were measured for an index of satisfaction. The sample mean is 28.8 and the sample standard deviation is 7.3. Use the given sample data to construct the 98 percent confidence interval for the population mean.

In a random sample of 500 college students, 23% say that they read or watch the news every day. Develop a 90% confidence interval for the proportion of all students who read or watch the news on a daily basis. Interpret your results. If you wanted to develop a 95% confidence interval with a margin of error of .01, how many students would need to be surveyed?