Improve Your Understanding of College Level Statistics Problems

Ten randomly selected people were asked how long they slept at night. The mean time was 7.1 hours, and the population variance was 0.6084 hour. Find the 99% confidence interval of the mean time. Assume the variable is normally distributed. In connection with the given data, explain the concept of the statement ‘the sample mean is significant or not significant at the given significance level’.

The resuts of a recent statewide test suggested that the proportion is 0.74. Using this estimate, what sample size is needed so that the confidence interval will have a margin of error of 0.07?
A sample of 7410 elementary school children is needed to obtain a 99.9% confidence interval with a margin of error of 0.07.

1. Given a random sample size of n = 900 from a bi- nomial probability distribution with P = 0.10 do the following:

   1. Find the probability that the number of successes is greater than 110.

   2. Find the probability that the number of successes is fewer than 53.

   3. Find the probability that the number of successes is between 55 and 120.

   4. With probability 0.10, the number of successes is fewer than how many?

   5. Withprobability0.08,thenumberofsuccessesis greater than how many?

Here are summary statistic for randomly selected weights of newborn girls: ${\displaystyle n=211,\bar{x}=33.7hg,s=6.8hg.}$ Construct a confidence intercal estimate of the mean. Use a 90% confidence level. Are these results very different from the confidence interval ${\displaystyle 32.1hg<\mu <34.3hg}$ with only 16 sample values, ${\displaystyle \bar{x}=33.2hg}$, and ${\displaystyle s=2.6hg}$?
What is the confidence interval for the population mean ${\displaystyle \mu }$?
Are the results between the two confidence intervals very different?
a) Yes, because the confidence interval limits are not similar.
b) No, because each confidence interval contains the mean of the other confidence interval.
c) No, because the confidence interval limits are similar
d) Yes, because one confidence interval does not contain the mean of the other confidence interval.

In order to estimate the mean 30-year fixed mortgage rate for a home loan in the United States, a random sample of 25 recent loans is taken. The average calculated from this sample is 6.95%. It can be assumed that 30-year fixed mortgage rates are normally distributed with a standard deviation of 0.6%. Compute 90% and 99% confidence intervals for the population mean 30-year fixed mortgage rate.

Let us assume that 1000 different random samples was selected from the same population and for each of them a confidence interval of the population mean was constructed (using exactly the same method) with confidence level of 96%. On average, how many of those 1000 different confidence intervals would contain the true value of the population mean?
What would be the answer if confidence level was 99%?

Consider all three-digit numbers that can be created from the digits 0-90-9 where the first and last digits must be odd and no digit can repeat. What is the probability of choosing a random number that starts with 11 from this group? Enter a fraction or round your answer to 44 decimal places, if necessary.

A recent survey of 7 social networking sites has a mean of 14.69 million visitors for a specifie month. The standard deviation was 4.4 millon. Find the 95% confidence interval of the true mean. Assume the variable is normally distributed. Round your answers to two decimal places.

Confidence intervals, again Several factors are involved in the creation of a confidence interval. Among them are the sample size, the level of confidence, and the margin of error. Which statements are true?
a) For a given sample size, reducing the margin of error will mean lower confidence.
b) For a certain confidence level, you can get a smaller margin of error by selecting a bigger sample.
c) For a fixed margin of error, smaller samples will mean lower confidence.
d) For a given confidence level, a sample 9 times as large will make a margin of error one third as big.

The average life span of a Golden Retriever is said to be 9.6 years. In a random sample of 15 Golden Retrievers, the following life spans were found:
6.8 10.2 9.5 13 9.9 7.0 8.3 8.9 11.2 10 9.8 13 9.8 9.5 8.7
At a 5% significance level, if we do a complete hypothesis test to test eh claim that the average life span of Golden Retrievers is 9.5, the p-value and conclusion are:
p-value = 0.8187, fail to reject the null hypothesis
p-value = 0.8153, fail to reject the null hypothesis
p-value = 0.8187, reject the null hypothesis
p-value = 0.8153, reject the null hypothesis

Suppose summation ${\displaystyle a_n}$ and summation ${\displaystyle b_n}$ are series with positive terms and summation ${\displaystyle b_n}$ is known to be divergent. If ${\displaystyle a_n<b_n}$ for all n, what can you say about summation ${\displaystyle a_n}$? Why?

The manufacturer of LED lights would like to include as part of their advertising the number of hours an LED is expected to work. A sample of 20 lights revealed the following number of hours worked.
$$\begin{array}{|cccccccccc|}\hline 2698& 2628& 2474& 2895& 2672& 2675& 2486& 2852& 2733& 2655\\ 2765& 2794& 2486& 2627& 2624& 2785& 2692& 2654& 2518& 2650\\ \hline\end{array}$$
Part 1: Draw normal probability plot for the above data and make comment.
Part 2: Assuming given data to be approximately normally distributed, Construct 90%, 95% and 99% confidence intervals for the mean number of hours that the LED lights of this brand has worked.
Part 3: Construct 90%, 95% and 99% confidence intervals for the standard deviation of the number of hours that the LED lights of this brand worked.

Which of the following are true? If false, explain briefly.
a) If the null hypothesis is true, youll

For a test of ${\displaystyle H_o:p=0.5}$, the z test statistic equals ${\displaystyle 1.52}$. Find the p-value for ${\displaystyle H_a:p>0.5}$.

An article regarding interracial dating and marriage recently appeared in a newspaper. Of the 1713 randomly selected adults, 301 identified themselves as Latinos, 325 identified themselves as blacks, 255 identified themselves as Asians, and 778 identified themselves as whites. Among Asians, 79% would welcome a white person into their families, 71% would welcome a Latino, and 66% would welcome a black person.
If you are using a Student's t-distribution, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)
Construct the 95% confidence intervals for the three Asian responses. (Round your answers to four decimal places.)
Construct the 95% confidence intervals for the three Asian responses. (Round your answers to four decimal places.)
a) Welcome a white peron.
b) welcome a black person.
c) welcome a latino

Give an example of each or state that the request is impossible. For any that are impossible, give a compelling argument for why that is the case.
a) A sequence with an infinite number of ones that does not converge to one.
b) A sequence with an infinite number of ones that converges to a limit not equal to one.
c) A divergent sequence such that for every ${\displaystyle n\in N}$ it is possible to find nn consecutive ones somewhere in the sequence.

For a test of Ho: p = 0.5, the z test statistic equals -1.74. Find the p-value for Ha: p < 0.5.
a. 0.0892
b. 0.0446
c. 0.0409
d. 0.9554
e. 0.0818

Determine (without solving the problem) an interval in which the solution of the given initial value problem is certain to exist. ${\displaystyle (t-3)y^{\prime }+\left(\ln t\right)y=2t,y\left(1\right)=2}$

In an experiment the sorption was measured for a variety of hazardous organic chloroalkane solvents. The sample measurements were; 1.18, 1.35, 0.57, 1.16, 1.12, 0.91, 0.83, 0.43 Assuming that measurements are normally distributed.
i) What is the mean of the sample for sorption of chloroalkane solvents.
ii) What is the standard error of the mean for the sample of organic chloroalkane solvents?
iii) What the margin of error of the 99% confidence intervals for the population mean of sorption of chloroalkane solvents.
iv) What is the lower limit for the 99% confidence intervals for the population mean of sorption of chloroalkane solvents.