Question 1

The population of lengths of aluminum-coated steel sheets is normally distributed with a mean of 30.05 inches and a standard deviation of 0.2 inches.What is the probability that a randomly selected sample of 4 sheets will have an average length of less than 29.9 inches long?

Accepted Answer

We can use the central limit theorem to approximate the distribution of sample means. The distribution of sample means will also be normally distributed with a mean of μ (30.05) and a standard deviation of σ/√n (0.2/√4 = 0.1).

We need to find the probability that a sample of 4 sheets will have a sample mean of less than 29.9 inches long.

z = (x̄ - μ) / (σ/√n) 
z = (29.9 - 30.05) / (0.1) 
z = -1.5

Using the standard normal distribution table, we can find the area to the left of z = -1.5, which is 0.0668. Therefore, the probability that a randomly selected sample of 4 sheets will have an average length of less than 29.9 inches long is 0.0668.

Question 2

A random sample of size 1,000 is taken from a population where p = .20.Find $\mathrm { P } ( \hat { p } < .22 )$

Accepted Answer

The sample proportion $\hat{p}$ is approximately normally distributed with mean $p = 0.20$ and standard deviation $\sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.20 \times 0.80}{1000}} = 0.0126$. We find the z-score for $\hat{p} = 0.22$ and use the standard normal distribution to find the probability: $z = \frac{0.22 - 0.20}{0.0126} \approx 1.5873$. The probability $P(Z < 1.5873)$ is approximately 0.9429.

Question 3

A random sample of size 1,000 is taken from a population where p = .20.Find $\mathrm { P } ( \hat { p } < .18 )$

Accepted Answer

In this case, the population proportion (p) is known and the sample size (n) is large enough (n=1000), so we can use the normal approximation to the binomial distribution. The mean of the sampling distribution of the sample proportion is equal to p=0.20, and the standard deviation is equal to sqrt(p*(1-p)/n) = sqrt(0.20*0.80/1000) = 0.0141. To find P(\hat{p} < 0.18), we standardize the sample proportion as (0.18 - 0.20)/0.0141 = -1.42 and look up the area to the left of -1.42 under the standard normal curve, which is 0.0793. Therefore, the answer is B.

Question 4

Find $\mathrm { P } ( 395.4 < \bar { X } < 404.6 )$ If the population mean = 400,?_x = 20,and n = 100.

Accepted Answer

The standard error of the mean is $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{100}} = 2$. The z-scores for 395.4 and 404.6 are $\frac{395.4 - 400}{2} = -2.3$ and $\frac{404.6 - 400}{2} = 2.3$, respectively. The probability $\mathrm{P}(395.4 < \bar{X} < 404.6)$ is the area between these z-scores, which is approximately 0.9786.

Question 5

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.Find $\mathrm { P } ( \bar { X } < 51.5 )$

Accepted Answer

The answer of A random sample of size 36 is...

Question 6

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.Find $\mathrm { P } ( \bar { X } > 50.5 )$

Accepted Answer

The population distribution is normal because the sample size is greater than 30. The mean of the population is given as μ = 50 and the standard deviation as σ = 5. The sample size is n = 36.

To find $\mathrm {P}(\bar {X} > 50.5 )$, we need to standardize the sample mean using the central limit theorem.

\begin{aligned} z &= \frac{\bar {X} - \mu}{\frac{\sigma}{\sqrt[]{n}}} \ &= \frac{50.5-50}{\frac{5}{\sqrt[]{36}}} \ &= 3.6 \end{aligned}

From the standard normal distribution table, we find that the area to the right of z = 3.6 is 0.0002. Hence,

\begin{aligned} \mathrm{P}(\bar {X} > 50.5) &= \mathrm{P}\left(z > 3.6\right) \ &= \mathrm{P}\left(z < -3.6\right) \ &= 0.0002 \end{aligned}

Therefore, the answer is (C).

Question 7

A random sample of size 1,000 is taken from a population where p = .20.Find $\mathrm { P } ( \hat { p } > 175 )$

Accepted Answer

The answer of A random sample of size 1,000 is...

Question 8

The population of lengths of aluminum-coated steel sheets is normally distributed with a mean of 30.05 inches and a standard deviation of 0.2 inches.A sample of four metal sheets is randomly selected from a batch.What is the probability that the average length of a sheet is between 30.25 and 30.35 inches long?

Accepted Answer

The answer of The population of lengths of aluminum-coated steel...

Question 9

A random sample of size 1,000 is taken from a population where p = .20.Find $\mathrm { P } ( \hat { p } > .21 )$

Accepted Answer

The answer of A random sample of size 1,000 is...

Question 10

The number of defectives in 10 different samples of 100 observations each is the following: 1,2,1,0,2,3,1,4,2,1.What is the estimate of the population proportion of defectives?

Accepted Answer

P ===.017

Question 11

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.What is $\sigma _ { \bar { X }}$
?

Accepted Answer

The standard deviation of the sample mean, $\sigma_{\bar{X}}$, is calculated using the formula $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size. Here, $\sigma = 5$ and $n = 36$, so $\sigma_{\bar{X}} = \frac{5}{\sqrt{36}} = \frac{5}{6} = 0.833$.

Question 12

Suppose that 60 percent of the voters in a particular region support a candidate.Find the probability that a sample of 1,000 voters would yield a sample proportion in favor of the candidate within 2 percentage points.

Accepted Answer

The answer of Suppose that 60 percent of the voters...

Question 13

A random sample of size 1,000 is taken from a population where p = .20.What is ${?} _{ \hat { p }} $
?

Accepted Answer

The answer of A random sample of size 1,000 is...

Question 14

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.What is μ_x?

Accepted Answer

The population mean is 50, so the expected value of the sample mean is also 50. The standard deviation of the sample mean is given by sigma/sqrt(n), where sigma is the population standard deviation and n is the sample size. Plugging in the given values, we get sigma/sqrt(36) = 5/sqrt(36) = 5/6. The standard error of the mean is the same as the standard deviation of the sample mean, so we have mu = 50 and s/sqrt(n) = 5/6. Therefore, the answer is 50.

Question 15

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.Find $\mathrm { P } ( \bar { X } > 49 )$

Accepted Answer

The Central Limit Theorem can be applied here since n > 30. Thus, the sample mean will be approximately normally distributed with mean = population mean = 50 and standard deviation = population standard deviation / sqrt(n) = 5 / sqrt(36) = 5/6. 
Now, we need to find the probability that the sample mean is greater than 49: 
z = (49-50) / (5/6) = -0.833 
Using the standard normal distribution table, the probability of getting a z-score less than -0.833 is 0.2019. 
Therefore, the probability of getting a z-score greater than -0.833 (or a sample mean greater than 49) is 1 - 0.2019 = 0.7981, which is 0.8849 when rounded to four decimal places. Hence the answer is C.

Question 16

A random sample of size 1,000 is taken from a population where p = .20.What is $\left( \sigma _ { \dot { p } } \right)$
?

Accepted Answer

The standard deviation of the sample proportion ($\sigma _{\dot{p}}$) is calculated using the formula $\sqrt{\frac{p(1-p)}{n}}$. Substituting the given values, $\sqrt{\frac{0.20 \times 0.80}{1000}} = 0.01265$.

Question 17

The number of defectives in 10 different samples of 50 observations each is the following: 5,1,1,2,3,3,1,4,2,3.What is the estimate of the population proportion of defectives?

Accepted Answer

P ===.05

Question 18

Packages of sugar bags for Sweeter Sugar Inc.have an average weight of 16 ounces and a standard deviation of 0.3 ounces.The weights of the sugar packages are normally distributed.What is the probability that 9 randomly selected packages will have a weight in excess of 16.025 ounces?

Accepted Answer

The answer of Packages of sugar bags for Sweeter Sugar...

Question 19

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.Find $\mathrm { P } ( \bar { X } < 48 )$

Accepted Answer

The answer of A random sample of size 36 is...

Question 20

Suppose that 60 percent of the voters in a particular region support a candidate.Find the probability that a sample of 1,000 voters would yield a sample proportion in favor of the candidate within 4 percentage points of the actual proportion.

Accepted Answer

σ_P == .0155 z = .04/.0155 = 2.58 P = .9951 − .0049 = .9902

Quiz 7: Sampling and Sampling Distributions

The population of lengths of aluminum-coated steel sheets is normally distributed with a mean of 30.05 inches and a standard deviation of 0.2 inches.What is the probability that a randomly selected sample of 4 sheets will have an average length of less than 29.9 inches long?

A random sample of size 1,000 is taken from a population where p = .20.Find $\mathrm { P } ( \hat { p } < .22 )$

A random sample of size 1,000 is taken from a population where p = .20.Find $\mathrm { P } ( \hat { p } < .18 )$

Find $\mathrm { P } ( 395.4 < \bar { X } < 404.6 )$ If the population mean = 400,?_x = 20,and n = 100.

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.Find $\mathrm { P } ( \bar { X } < 51.5 )$

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.Find $\mathrm { P } ( \bar { X } > 50.5 )$

A random sample of size 1,000 is taken from a population where p = .20.Find $\mathrm { P } ( \hat { p } > 175 )$

A random sample of size 1,000 is taken from a population where p = .20.Find $\mathrm { P } ( \hat { p } > .21 )$

The number of defectives in 10 different samples of 100 observations each is the following: 1,2,1,0,2,3,1,4,2,1.What is the estimate of the population proportion of defectives?

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.What is $\sigma _ { \bar { X }}$ ?

Suppose that 60 percent of the voters in a particular region support a candidate.Find the probability that a sample of 1,000 voters would yield a sample proportion in favor of the candidate within 2 percentage points.

A random sample of size 1,000 is taken from a population where p = .20.What is ${?} _{ \hat { p }}$ ?

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.What is μ_x?

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.Find $\mathrm { P } ( \bar { X } > 49 )$

A random sample of size 1,000 is taken from a population where p = .20.What is $\left( \sigma _ { \dot { p } } \right)$ ?

The number of defectives in 10 different samples of 50 observations each is the following: 5,1,1,2,3,3,1,4,2,3.What is the estimate of the population proportion of defectives?

Packages of sugar bags for Sweeter Sugar Inc.have an average weight of 16 ounces and a standard deviation of 0.3 ounces.The weights of the sugar packages are normally distributed.What is the probability that 9 randomly selected packages will have a weight in excess of 16.025 ounces?

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.Find $\mathrm { P } ( \bar { X } < 48 )$

Suppose that 60 percent of the voters in a particular region support a candidate.Find the probability that a sample of 1,000 voters would yield a sample proportion in favor of the candidate within 4 percentage points of the actual proportion.

Quiz 7: Sampling and Sampling Distributions

The population of lengths of aluminum-coated steel sheets is normally distributed with a mean of 30.05 inches and a standard deviation of 0.2 inches.What is the probability that a randomly selected sample of 4 sheets will have an average length of less than 29.9 inches long?

A random sample of size 1,000 is taken from a population where p = .20.Find P(p^<.22) \mathrm { P } ( \hat { p } < .22 ) P(p^​<.22)

A random sample of size 1,000 is taken from a population where p = .20.Find P(p^<.18) \mathrm { P } ( \hat { p } < .18 ) P(p^​<.18)

Find P(395.4<Xˉ<404.6) \mathrm { P } ( 395.4 < \bar { X } < 404.6 ) P(395.4<Xˉ<404.6) If the population mean = 400,?x = 20,and n = 100.

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.Find P(Xˉ<51.5) \mathrm { P } ( \bar { X } < 51.5 ) P(Xˉ<51.5)

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.Find P(Xˉ>50.5) \mathrm { P } ( \bar { X } > 50.5 ) P(Xˉ>50.5)

A random sample of size 1,000 is taken from a population where p = .20.Find P(p^>175) \mathrm { P } ( \hat { p } > 175 ) P(p^​>175)

A random sample of size 1,000 is taken from a population where p = .20.Find P(p^>.21) \mathrm { P } ( \hat { p } > .21 ) P(p^​>.21)

The number of defectives in 10 different samples of 100 observations each is the following: 1,2,1,0,2,3,1,4,2,1.What is the estimate of the population proportion of defectives?

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.What is σXˉ\sigma _ { \bar { X }}σXˉ​ ?

Suppose that 60 percent of the voters in a particular region support a candidate.Find the probability that a sample of 1,000 voters would yield a sample proportion in favor of the candidate within 2 percentage points.

A random sample of size 1,000 is taken from a population where p = .20.What is ?p^{?} _{ \hat { p }} ?p^​​ ?

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.What is μx?

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.Find P(Xˉ>49) \mathrm { P } ( \bar { X } > 49 ) P(Xˉ>49)

A random sample of size 1,000 is taken from a population where p = .20.What is (σp˙) \left( \sigma _ { \dot { p } } \right) (σp˙​​) ?

The number of defectives in 10 different samples of 50 observations each is the following: 5,1,1,2,3,3,1,4,2,3.What is the estimate of the population proportion of defectives?

Packages of sugar bags for Sweeter Sugar Inc.have an average weight of 16 ounces and a standard deviation of 0.3 ounces.The weights of the sugar packages are normally distributed.What is the probability that 9 randomly selected packages will have a weight in excess of 16.025 ounces?

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.Find P(Xˉ<48) \mathrm { P } ( \bar { X } < 48 ) P(Xˉ<48)

Suppose that 60 percent of the voters in a particular region support a candidate.Find the probability that a sample of 1,000 voters would yield a sample proportion in favor of the candidate within 4 percentage points of the actual proportion.

A random sample of size 1,000 is taken from a population where p = .20.Find $\mathrm { P } ( \hat { p } < .22 )$

A random sample of size 1,000 is taken from a population where p = .20.Find $\mathrm { P } ( \hat { p } < .18 )$

Find $\mathrm { P } ( 395.4 < \bar { X } < 404.6 )$ If the population mean = 400,?_x = 20,and n = 100.

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.Find $\mathrm { P } ( \bar { X } < 51.5 )$

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.Find $\mathrm { P } ( \bar { X } > 50.5 )$

A random sample of size 1,000 is taken from a population where p = .20.Find $\mathrm { P } ( \hat { p } > 175 )$

A random sample of size 1,000 is taken from a population where p = .20.Find $\mathrm { P } ( \hat { p } > .21 )$

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.What is $\sigma _ { \bar { X }}$ ?

A random sample of size 1,000 is taken from a population where p = .20.What is ${?} _{ \hat { p }}$ ?

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.What is μ_x?

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.Find $\mathrm { P } ( \bar { X } > 49 )$

A random sample of size 1,000 is taken from a population where p = .20.What is $\left( \sigma _ { \dot { p } } \right)$ ?

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5.Find $\mathrm { P } ( \bar { X } < 48 )$