The mean height of adult males in the U.S is about 68.8" with a variance of 11.56". Suppose the mean height of a sample of 31 mentally disabled males was found to be 67.3". Researchers want to know if the height of mentally disabled males differs from the population. What can be concluded with α = 0.01?

Compute the appropriate test statistic(s) to make a decision about H₀.
critical value =  

If appropriate, compute the CI. If not appropriate, input "na" for both spaces below.
[   ,   ]

e) Compute the corresponding effect size(s) and indicate magnitude(s).
d =   ;   ---Select--- na trivial effect small effect medium effect large effect

On a planet far far away from Earth, IQ of the ruling species is normally distributed with a mean of 105 and a standard deviation of 14. Suppose one individual is randomly chosen. Let X = IQ of an individual.

a. What is the distribution of X? X ~ N(,)

b. Find the probability that a randomly selected person's IQ is over 97. Round your answer to 4 decimal places.

c. A school offers special services for all children in the bottom 4% for IQ scores. What is the highest IQ score a child can have and still receive special services? Round your answer to 2 decimal places.

d. Find the Inter Quartile Range (IQR) for IQ scores. Round your answers to 2 decimal places.

Q1:

Q3:

IQR:

Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us.

(a)

• Suppose n = 25 and p = 0.28.

(For each answer, enter a number. Use 2 decimal places.)
n·p = ?
n·q = ?

Can we approximate p̂ by a normal distribution? Why? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, p̂ _____ be approximated by a normal random variable because _____ _____.

first blank

Yes or No    

second blank

can or cannot    

third blank

n·p exceedsn·q exceeds
n·p does not exceed
n·p and n·q do not exceed
both n·p and n·q exceed
n·q does not exceed

fourth blank (Enter an exact number.)
?

What are the values of μ_p̂ and σ_p̂? (For each answer, enter a number. Use 3 decimal places.)
μ_p̂ = mu sub p hat = ?

σ_p̂ = sigma sub p hat = ?

(b)

Suppose

• n = 25 and p = 0.15.

Can we safely approximate p̂ by a normal distribution? Why or why not? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, p̂ _____ be approximated by a normal random variable because _____ _____.

first blank

Yes or No    

second blank

can or cannot    

third blank

n·p exceedsn·q exceeds
n·p does not exceed
n·p and n·q do not exceed
both n·p and n·q exceed
n·q does not exceed

fourth blank (Enter an exact number.)
?

(c)

Suppose

• n = 48 and p = 0.40.

(For each answer, enter a number. Use 2 decimal places.)
n·p = ?
n·q = ?

Can we approximate p̂ by a normal distribution? Why? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, p̂ _____ be approximated by a normal random variable because _____ _____.

first blank

Yesor No    

second blank

can or cannot    

third blank

n·p exceedsn·q exceeds
n·p does not exceed
n·p and n·q do not exceed
both n·p and n·q exceed
n·q does not exceed

fourth blank (Enter an exact number.)
?

What are the values of μ_p̂ and σ_p̂? (For each answer, enter a number. Use 3 decimal places.)
μ_p̂ = mu sub p hat =

σ_p̂ = sigma sub p hat =

For the data set

(a) Find the 76th percentile.

(b) Find the 42nd percentile.

(c) Find the 16th percentile.

(d) Find the 65th percentile.

Suppose x has a normal distribution with a mean of 79 and a variance of 441.00. If a sample of 15 were randomly drawn from the population, find the probability of   mu hat   for each of the following situations.

a) less than 77: probability =

b) greater than 83: probability =

c) in between 65 and 76: probability =

d) in between 76 and 94: probability =

Suppose 31 pregnant women are sampled who smoke an average of 22 cigarettes per day with a variance of 144.00.

a) What is the probability that the pregnant women will smoke an average of 20 cigarettes or more? probability =

b) What is the probability that the pregnant women will smoke an average of 21 cigarettes or less? probability =

c) What is the probability that the pregnant women will smoke an average of 18 to 24 cigarettes? probability =

d) What is the probability that the pregnant women will smoke an average of 23 to 26 cigarettes? probability =

Note: Do NOT input probability responses as percentages; e.g., do NOT input 0.9194 as 91.94.

A researcher wishes to test the effects of excerise on the ability to complete a basic skills test. He designs a pre-test and post-test to give to each participant. You believe that there was an increase in the scores. You believe the population of the differences is normally distributed, but you do not know the standard deviation. When calculating difference use Post-test minus Pre-test. pre-test post-test 52 60 60 55 44 49 92 94 84 76 55 65 64 58 67 66 53 59 99 99 75 79 77 82 Which of the following are the correct hypotheses? H 0 : μ d ≥ 0 H 0 : μ d ≥ 0 H A : μ d < 0 H A : μ d < 0 (claim) H 0 : μ d ≤ 0 H 0 : μ d ≤ 0 H A : μ d > 0 H A : μ d > 0 (claim) H 0 : μ d = 0 H 0 : μ d = 0 H A : μ d ≠ 0 H A : μ d ≠ 0 (claim) Correct

Given that α α is 0.10 the critical value is 1.363

The test statistic is: Incorrect(round to 3 places)

The p-value is: Incorrect(round to 3 places)

In an effort to improve the mathematical skills of 18 students, a teacher provides a weekly 1-hour tutoring session. A pre-test is given before the sessions and a post-test is given after. The results are shown here. Test the claim that there was an increase in the scores. at αα=0.01. You believe that the population is normally distributed, but you do not know the standard deviation. When calculating difference use Post-test minus Pre-test.

Which of the following are the correct hypotheses?

• H0:μd≥0H0:μd≥0
  HA:μd<0HA:μd<0(claim)
• H0:μd≤0H0:μd≤0
  HA:μd>0HA:μd>0(claim)
• H0:μd=0H0:μd=0
  HA:μd≠0HA:μd≠0(claim)

Given that αα is 0.01 the critical value is 2.567
The test statistic is: (round to 3 places)
The p-value is: (round to 3 places)

Suppose that the national average for the math portion of the College Board's SAT is 518. The College Board periodically rescales the test scores such that the standard deviation is approximately 50. Answer the following questions using a bell-shaped distribution and the empirical rule for the math test scores. If required, round your answers to two decimal places. (a) What percentage of students have an SAT math score greater than 568? % (b) What percentage of students have an SAT math score greater than 618? % (c) What percentage of students have an SAT math score between 468 and 518? % (d) What is the z-score for student with an SAT math score of 620? (e) What is the z-score for a student with an SAT math score of 405?

Your claim results in the following alternative hypothesis:
H_a : μ ≠≠ 166
which you test at a significance level of α=.02α=.02.

Find the positive critical value, to three decimal places.

z_α/2 =

You are performing a right-tailed t-test with a sample size of 5

If α=.005α=.005, find the critical value, to two decimal places.

You are performing a two-tailed test.

If α=.06α=.06, find the positive critical value, to three decimal places.

z_α/2 =

Testing:

H0:μ=7.83H0:μ=7.83
H1:μ>7.83H1:μ>7.83

Your sample consists of 23 subjects, with a mean of 8.3 and a sample standard deviation (s) of 4.14.

Calculate the test statistic, rounded to 2 decimal places.

t=t=

You are performing a two-tailed z-test

If α=0.1α=0.1, find the positive critical value, to two decimal places.

Using R Studio

Now, set the seed to 348 with `set.seed()`. Then take a sample of size 10,000 from a normal distribution with a mean of 82 and a standard deviation of 11.

(a) Using sum() on a logical vector, how many draws are less than 60? Using mean() on a logical vector, what proportion of the total draws is that? How far is your answer from pnorm() in 1.1 above?

```{R}
set.seed(348)
x=rnorm(10000,82,11)
sum(ifelse(x<60,1,0))

mean(ifelse(x<60,1,0))

pnorm(60,82,11)

Using sum() function there are 128 draws that are less than 60 and using the mean() function 0.0281 is the porportion of total draws. From these outputs we can say that the answer is quite close to the pnorm() value that has been calculated.

(b) What proportion of your sample is greater than 110 or less than 54?

(c) Why are your answers close to what you got above? Why are they not exactly the same?

(d) Using ggplot2, make a histogram of your sample. Set y=..density.. inside aes(). Overlay a normal distribution with stat_function(aes(samp), fun=dnorm, args=list(82,11)). Using geom_vline(xintercept=), add dashed vertical lines corresponding to the 2.5th and the 97.5th percentile of the sample

In the Focus Problem at the beginning of this chapter, a study was described comparing the hatch ratios of wood duck nesting boxes. Group I nesting boxes were well separated from each other and well hidden by available brush. There were a total of 469 eggs in group I boxes, of which a field count showed about 262 hatched. Group II nesting boxes were placed in highly visible locations and grouped closely together. There were a total of 804 eggs in group II boxes, of which a field count showed about 276 hatched.

(a) Find a point estimate p̂<sub>1</sub> for p<sub>1</sub>, the proportion of eggs that hatch in group I nest box placements. (Round your answer to three decimal places.)
p̂<sub>1</sub> =

Find a 90% confidence interval for p<sub>1</sub>. (Round your answers to three decimal places.)

(b) Find a point estimate p̂<sub>2</sub> for p<sub>2</sub>, the proportion of eggs that hatch in group II nest box placements. (Round your answer to three decimal places.)
p̂<sub>2</sub> =

Find a 90% confidence interval for p<sub>2</sub>. (Round your answers to three decimal places.)

(c) Find a 90% confidence interval for p<sub>1</sub> − p<sub>2</sub>. (Round your answers to three decimal places.)

Does the interval indicate that the proportion of eggs hatched from group I nest boxes is higher than, lower than, or equal to the proportion of eggs hatched from group II nest boxes?

Because the interval contains only positive numbers, we can say that a higher proportion of eggs hatched in well-separated and well-hidden nesting boxes.Because the interval contains only negative numbers, we can say that a higher proportion of eggs hatched in highly visible, closely grouped nesting boxes.    We can not make any conclusions using this confidence interval.Because the interval contains both positive and negative numbers, we can not say that a higher proportion of eggs hatched in well-separated and well-hidden nesting boxes.

(d) What conclusions about placement of nest boxes can be drawn? In the article discussed in the Focus Problem, additional concerns are raised about the higher cost of placing and maintaining group I nest box placements. Also at issue is the cost efficiency per successful wood duck hatch.

A greater proportion of wood duck eggs hatch if the eggs are laid in highly visible, closely grouped nesting boxes.No conclusion can be made.    A greater proportion of wood duck eggs hatch if the eggs are laid in well-separated, well-hidden nesting boxes.The eggs hatch equally well in both conditions.

Edwards Manufacturing Company purchases two component parts from three different suppliers. The suppliers have limited capacity, and no one supplier can meet all the company’s needs. In addition, the suppliers charge different prices for the components. Component price data (in price per unit) are as follows:

Each supplier has a limited capacity in terms of the total number of components it can supply. However, as long as Edwards provides sufficient advance orders, each supplier can devote its capacity to component 1, component 2, or any combination of the two components, if the total number of units ordered is within its capacity. Supplier capacities are as follows:

If the Edwards production plan for the next period includes 1025 units of component 1 and 825 units of component 2, what purchases do you recommend? That is, how many units of each component should be ordered from each supplier? Round your answers to the nearest whole number. If your answer is zero, enter "0".

What is the total purchase cost for the components? Round your answer to the nearest dollar.

$__________

The distribution of weights for 12 month old baby girls in the US is approximately normal with mean u = 21 pounds and standard deviation of 2.2 pounds.

a) if a 12 month old girl weighs 23.2 pounds, approximately what weight percentile is she in?

b) if a 12 month old girl is in the 16th percentile in weight, estimate her weight.

c) Estimate the weight of t 12 month old girl who is in the 25th percentile by weight.

d) Estimate the weight of a 12 month old girl who is in the 75th percentile by weight.

Let X represent the weight of the students at a university. Suppose X has a mean of 75 kg and a standard deviation of 10 kg. Among 100 such randomly selected students from this university, what is the approximate probability that the average weight of this sample (X100) lies between

(a) 74 and 75 kg

(b) greater than 76 kg

(c) less than 73 kg

Assume that the sample size(N) is large enough for the CLT (Central Limit Theorem) to be applicable.