There is a test which is normally distributed with a mean of 30 and a standard deviation of 4. Find the probability that a sample of 36 scores will have a mean at most 31.5.

The mean weight of an adult is 60 kilograms with a variance of 100

If 118 adults are randomly selected, what is the probability that the sample mean would be greater than 62.2 kilograms? Round your answer to four decimal places.

A worker is given monthly tests (new test generated each month) to assess his ability to be effective in the workplace. However this worker got into an accident recently. The employer said that "the accident would change them, and the employer believe that the worker is a completely different person before versus after the accident. The employer would like to determine if the worker's performance has significantly changed since the accident has occured.

alpha = 5%
before = sample 1

(a) Find the p-values of the normality tests

p-value (before) =

p-value (after) =

(b) State the hypotheses:

(c) Find the test statistic for this test =

(d) Determine the ?P-value of your statistical test =

(e) At a=0.05, the data indicates you will [reject / fail to reject] null hypothesis. You can say that the test scores before and after the accident is _______ after the accident.

Data:

The manufacturer of a certain electronic component claims that they are designed to last just slightly more than 4 years because they believe that customers typically replace their device before then. Based on information provided by the company, the components should last a mean of 4.24 years with a standard deviation of 0.45 years. For this scenario, assume the lifespans of this component follow a normal distribution.

1. The company offers a warranty for this component that allows customers to return for a refund it if it fails in less than 4 years. What is the probability that a randomly chosen component will last less than 4 years?

2.The company considers a component to be “successful” if it lasts longer than the warranty period before failing. They estimate that about 70.3% of components last more than 4 years. They find a random group of 10 components that were sold and count the number of them which were “successful,” lasting more than 4 years.

What is the probability that at least 8 of these components will last more than 4 years?

A researcher would like to predict the dependent variable Y from the two independent variables X1 and X2 for a sample of N=13 subjects. Use multiple linear regression to calculate the coefficient of multiple determination and test statistics to assess the significance of the regression model and partial slopes. Use a significance level α=0.02.

R2=
F=
P-value for overall model =

t1=
for b1, P-value =
t2=
for b2, P-value =

What is your conclusion for the overall regression model (also called the omnibus test)?

• The overall regression model is statistically significant at α=0.02.
• The overall regression model is not statistically significant at α=0.02.

Which of the regression coefficients are statistically different from zero?

• neither regression coefficient is statistically significant
• the slope for the first variable b1 is the only statistically significant coefficient
• the slope for the second variable b2 is the only statistically significant coefficient
• both regression coefficients are statistically significant

Determine the 95​% confidence interval estimate for the population mean of a normal distribution given n=144​, sigma=105​, and x overbar=1,200. The 95​% confidence interval for the population mean is from to . ​(Round to two decimal places as needed. Use ascending​ order.

Answer the following questions with clear explanation.

a. Explain intuitively what does population mean and variance tells you about the distribution of a random variable?

b. Can you think of a reason why do we mostly only care about mean and variance a random variable since the distribution function could involve more than the two parameters?

c. Explain why we could use a sample data to draw conclusions about a population parameter?

d. Is p-value of a hypothesis test a random variable, why? e. If you reject your null hypothesis, you have proved that the null hypothesis is wrong. Is this a correct statement, why?

Part 1 The world's smallest mammal is the bumblebee bat. The mean weight of 60 randomly selected bumblebee bats is 1.659 grams, with a standard deviation of 0.264 grams. Find a 99.9% confidence interval for the mean weight of all bumblebee bats at the following confidence levels (two places after decimal): (WebAssign will check your answer for the correct number of significant figures. , WebAssign will check your answer for the correct number of significant figures. ) Find a 99% confidence interval for the mean weight of all bumblebee bats at the following confidence levels (two places after decimal): (WebAssign will check your answer for the correct number of significant figures. , WebAssign will check your answer for the correct number of significant figures. ) Find a 95% confidence interval for the mean weight of all bumblebee bats at the following confidence levels (two places after decimal): (Web Assign will check your answer for the correct number of significant figures. , WebAssign will check your answer for the correct number of significant figures. ) Find an 80% confidence interval for the mean weight of all bumblebee bats at the following confidence levels (two places after decimal): (WebAssign will check your answer for the correct number of significant figures. , WebAssign will check your answer for the correct number of significant figures. ) Part 2 Dr. Clifford Jones claims that the mean weight of bumblebee bats is 1.8 grams. We are interested in whether it's less than he claims. Using the t-distribution technique, is there evidence that bumblebee bats weight less on average than he claims, at each of the following levels? The degrees of freedom are df = The test-statistic t = (positive version of t, three places after decimal) = WebAssign will check your answer for the correct number of significant figures. Is there evidence at the 10% level that bumblebee bats weigh less on average than he claims?

A survey of 750 likely voters in Ohio was conducted by the Rasmussen Poll just prior to the general election, The state of the economy was thought to be an important determinant of how people would vote. Among other things, the survey found that 165 of the respondents rated the economy as good or excellent, 315 rated the economy as poor (the rest of the respondents were in between these two categories).

a. (10p) Develop a point estimate of the proportion of likely voters in Ohio who rated the economy as good or excellent.

b. (20p) Construct a 90% confidence interval for the proportion of likely voters in Ohio who rated the economy as good or excellent.

c. (20p) Construct a 90% confidence interval for the proportion of likely voters in Ohio who rated the economy as poor.

A) state the complete Central Limit Theorem (CLT)

B) explain why we need the theoretical idea of sampling distributions in a hypothesis test even though we only take one sample to decide between the hypothesis.

C) relate each part of the formula r= X-Mean₀ / S/ square root of n

D) Explain what type 1 and type 2 errors are

E) explain how it is possible to conduct the correct test flawlessly using a simple random sample of sufficient size and still commit a Type 1 and type 2 error. Use the idea of a sampling distribution in your explanation.

1. What is a probability distribution? Provide an example and discuss its basic features.
2. What do we mean when we say two events are independent? please Explain

Suppose you flip one million coins. Would you be surprised to find that 501,000 landed heads? Would you be surprised to find that 510,000 landed heads? Explain.

Say we have a continuous random variable X with density function f(x)=c(1+x3) (where c is a constant)with support SX =[0,3].

a.) What value of c will make f(x) a valid probability density function.

b. )What is the probability that X=2? What is the probability that X is greater than 2?

Now say we have an infinite sequence of independent random variables Xi (that is to say X1, X2, X3, ....) with density f(x) stated earlier.

c. What is the probability that the first random variable/trial to be greater than 2 is on the 10 trial (first 9 trials are less than 2 and the 10 trial is greater than 2) ?

d. What is the probability that it will take less than 10 random variables/trials before we see a trial that is greater than 2?

A survey is conducted on 700 Californians older than 30 years of age. The study wants to obtain inference on the relationship between years of education and yearly income in dollars. The response variable is income and the explanatory variable is years of education.
A simple linear regression model is fit, and the output from R is below.

lm(formula = Income ~ Education, data = CA)

Coefficients:
Estimate Std. Error t value Pr(>|t|)

(Intercept) 25200.20 1488.94 16.93 3.08e-10 ***

Education 2905.35 112.61 25.80 1.49e-12 ***

a)Write out the estimated linear equation. What is the estimated expected income of a Californian that has 12 years of education (high school level)?

b)Does the intercept have a useful interpretation in this study? Why or why not.

c)Interpret the slope estimate in context of the model. Now say you have two people, where one has 4 years more education than the other. What is the estimated difference in expected income?

d)The p-value to test the null hypothesis that the slope on Education is 0 (H0 : β1 = 0 vs Ha : β1 ̸= 0), is approximately 0. What can you say about Education being a significant explanatory variable or covariate when explaining Income?

A random sample of size n = 130 is taken from a population with a population proportion p = 0.58. (You may find it useful to reference the z table.)

a. Calculate the expected value and the standard error for the sampling distribution of the sample proportion. (Round "expected value" to 2 decimal places and "standard error" to 4 decimal places.)

b. What is the probability that the sample proportion is between 0.50 and 0.70? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.)

c. What is the probability that the sample proportion is less than 0.50? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.)