Question

1. A statistic is an unbiased estimator of a population parameter. This means that ...

A. ... the population mean is equal to the mean of the parameter.

B. ... the population is normal and the population mean is equal to the statistic.

C. ... the population is normal, the population mean is equal to the statistic mean, and the population standard deviation is equal to the parameter standard deviation.

D. ... the statistic mean is equal to the parameter.

2. An online survey was conducted to study cyberbullying in the UK. It was reported that 62% of young people had received nasty private messages on a social network. You randomly sample four young people from the network and ask them if they received nasty messages on the network. Find the probability that the number who received these messages is three.

A. .85 B. .36 C. .74 D. .18

3. The number of particles large enough to initiate a crack on an aircraft wing follows a Poisson distribution with mean 0.5 per square centimeter. Find the probability that a 100 square centimeter section has more than 60 of these particles.

A. .270 B. .207 C. .072 D. .027

6. In a survey of 1430 undergraduate students, 1087 reported that they had one or more credit cards. Give a 95% confidence interval for the proportion of all college students who have at least one credit card.

A. 73% to 79% B. 70% to 82% C. 65% to 60% D. 74% to 78%

7. A study compared the proportions of young women and men who use Instagram. Out of 537 women, 328 used Instagram, whereas out of 532 men, 234 used the service. Find a 95% confidence interval for the difference between the proportions of men and women who use Instagram.

A. .09 to .25 B. .10 to .24 C. .11 to .23 D. .12 to .22

8. As a result of the computation of a confidence interval with a 95% confidence level, it is reported that the proportion of American families that own more than one car is 12.4% with a margin of error of 2%. In light of this, we are 95% confident that the largest the actual proportion of such families can be is:

A. 14.4% B. 12.6% C. 12.42% D. 16.4%

Answer 1

## 1.) A statistic is an unbiased estimator of a population parameter. This means that ...

Answer : D. ... the statistic mean is equal to the parameter.

( sample mean is unbiased estimator of the population mean )  

correct Optionis D

## 2)  An online survey was conducted to study cyberbullying in the UK. It was reported that 62% of young people had received nasty private messages on a social network. You randomly sample four young people from the network and ask them if they received nasty messages on the network. Find the probability that the number who received these messages is three.

Answer : correct Option is B : 0.36

we can use binomial distribution : n = 4 , p = 0.62

p (X=3) = 4C3 * ( 0.62) ^3 * (1- 0.62) ^1

= 0.3622

## 3)   The number of particles large enough to initiate a crack on an aircraft wing follows a Poisson distribution with mean 0.5 per square centimeter. Find the probability that a 100 square centimeter section has more than 60 of these particles.

Answer : correct option is : C = 0.072

p ( x > 60 ) ?

we can use approximation of the poisson to normal distribution :  

here poisson paramater = 0.5 = lamda

mean = n * lamda = 100 * 0.5 = 50

and variance = n* lamda = 50

because mean = variance for poisson distribution .

p ( x > 60 ) we know that : z = (x - mean) / standard devation follows normal distribution ;

p (z > (60 - 50 ) / 7.07 ) =

p ( z > 1.41) = 1 - p (z< 1.41)

now use z score table : p ( z > 1.41) = 1 - p (z< 1.41)  

= 1 - 0.9207 = 0.0793

apprixiimate = 0.072

## 6) In a survey of 1430 undergraduate students, 1087 reported that they had one or more credit cards. Give a 95% confidence interval for the proportion of all college students who have at least one credit card.

Answer : Coorect option is : D ) 74% to 78%

here we have n = 1430 and x = 1087

sample proportion = phat = x /n = 0.7601

95 % confidence interval for population proportion : p = ( phat  ± critical value * SQRT (phat * (1 - phat) / n))

critical value = ± 1.96 ( from table )

p = ( 0.7601  ± 1.96 *  SQRT ( 0.7601  * (1 - 0.7601) / 1430 ))

p = ( 0.7601  ± 0.0221)

p = ( 0.7379 , 0.7822)

lower limit = 0.74 and upper limit = 0.78 ie

lower limit = 74 % and upper limit = 78 %  

## 7.) A study compared the proportions of young women and men who use Instagram. Out of 537 women, 328 used Instagram, whereas out of 532 men, 234 used the service. Find a 95% confidence interval for the difference between the proportions of men and women who use Instagram.

Answer : correct option is : C. ( 11 to .23)

conisder for women = x1

n1 = sample size for women = 537 , x1 = 328 , sample proportion for women = p1hat = x1 / n1

= ( 328 / 537 ) = 0.61

conisder for men = x2

n2 = sample size for men = 532 , x2 = 234 , sample proportion for men  = p2hat = x2 / n2

= ( 234 / 532 ) = 0.439

95 % confidence interval for diffrence between population proportion :

( p1 - p2 ) = [ ( p1hat - p2hat)   ± critical value * SQRT ( (p1hat * (1 - p1hat) / n1) + (p2hat * (1 - p2hat) / n2))]

critical value = ± 1.96 ( from table )

= [ ( 0.610 - 0.439 )  ± 1.96 *  SQRT ( (0.61 * (1 - 0.61) / 537) + ( 0.439 * (1 - 0.439) / 532))]

= (0.171   ± 1.96 * 0.0300 )

= (0.171   ± 0.0589 )

lower limit = 0.1121 ie approximately = 0.11

and upper limit = 0.2299 ie approximately = 0.23  

## 8. ) As a result of the computation of a confidence interval with a 95% confidence level, it is reported that the proportion of American families that own more than one car is 12.4% with a margin of error of 2%. In light of this, we are 95% confident that the largest the actual proportion of such families can be is:

Answer : correct option is ; C : 12 .42