Consider the marks of all 1st-year students on a statistics test. If the marks have a normal distribution with a mean of 72 and a standard deviation of 9, then the probability that a random sample of 10 students from this group have a sample mean between 71 and 73 is?

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asked 2021-01-31

factor in determining the usefulness of an examination as a measure of demonstrated ability is the amount of spread that occurs in the grades. If the spread or variation of examination scores is very small, it usually means that the examination was either too hard or too easy. However, if the variance of scores is moderately large, then there is a definite difference in scores between "better," "average," and "poorer" students. A group of attorneys in a Midwest state has been given the task of making up this year's bar examination for the state. The examination has 500 total possible points, and from the history of past examinations, it is known that a standard deviation of around 60 points is desirable. Of course, too large or too small a standard deviation is not good. The attorneys want to test their examination to see how good it is. A preliminary version of the examination (with slight modifications to protect the integrity of the real examination) is given to a random sample of 20 newly graduated law students. Their scores give a sample standard deviation of 70 points. Using a 0.01 level of significance, test the claim that the population standard deviation for the new examination is 60 against the claim that the population standard deviation is different from 60.
(a) What is the level of significance?
State the null and alternate hypotheses.
$H_{0}:\sigma=60,\ H_{1}:\sigma\ <\ 60H_{0}:\sigma\ >\ 60,\ H_{1}:\sigma=60H_{0}:\sigma=60,\ H_{1}:\sigma\ >\ 60H_{0}:\sigma=60,\ H_{1}:\sigma\ \neq\ 60$
(b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.)
What are the degrees of freedom?
What assumptions are you making about the original distribution?
We assume a binomial population distribution.We assume a exponential population distribution. We assume a normal population distribution.We assume a uniform population distribution.

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asked 2021-02-23

1. A researcher is interested in finding a 98% confidence interval for the mean number of times per day that college students text. The study included 144 students who averaged 44.7 texts per day. The standard deviation was 16.5 texts. a. To compute the confidence interval use a ? z t distribution. b. With 98% confidence the population mean number of texts per day is between and texts. c. If many groups of 144 randomly selected members are studied, then a different confidence interval would be produced from each group. About percent of these confidence intervals will contain the true population number of texts per day and about percent will not contain the true population mean number of texts per day. 2. You want to obtain a sample to estimate how much parents spend on their kids birthday parties. Based on previous study, you believe the population standard deviation is approximately $\displaystyle\sigma={40.4}$ dollars. You would like to be 90% confident that your estimate is within 1.5 dollar(s) of average spending on the birthday parties. How many parents do you have to sample? n = 3. You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the population standard deviation is approximately $\displaystyle\sigma={57.5}$. You would like to be 95% confident that your estimate is within 0.1 of the true population mean. How large of a sample size is required?

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asked 2020-11-09

A researcher is interested in finding a $90\%$ confidence interval for the mean number minutes students are concentrating on their professor during a one hour statistics lecture. The study included 117 students who averaged 40.9 minutes concentrating on their professor during the hour lecture. The standard deviation was 11.8 minutes. Round answers to 3 decimal places where possible.
a.
To compute the confidence interval use a ? distribution.
b.
With $90\%$ confidence the population mean minutes of concentration is between ____ and ____ minutes.
c.
If many groups of 117 randomly selected students are studied, then a different confidence interval would be produced from each group. About ____ percent of these confidence intervals will contain the true population mean minutes of concentration and about ____ percent will not contain the true population mean number of minutes of concentration.

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asked 2020-12-24

True or False
1.The goal of descriptive statistics is to simplify, summarize, and organize data.
2.A summary value, usually numerical, that describes a sample is called a parameter.
3.A researcher records the average age for a group of 25 preschool children selected to participate in a research study. The average age is an example of a statistic.
4.The median is the most commonly used measure of central tendency.
5.The mode is the best way to measure central tendency for data from a nominal scale of measurement.
6.A distribution of scores and a mean of 55 and a standard deviation of 4. The variance for this distribution is 16.
7.In a distribution with a mean of M = 36 and a standard deviation of SD = 8, a score of 40 would be considered an extreme value.
8.In a distribution with a mean of M = 76 and a standard deviation of SD = 7, a score of 91 would be considered an extreme value.
9.A negative correlation means that as the X values decrease, the Y values also tend to decrease.
10.The goal of a hypothesis test is to demonstrate that the patterns observed in the sample data represent real patterns in the population and are not simply due to chance or sampling error.

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asked 2020-10-23

1. Find each of the requested values for a population with a mean of $? = 40$, and a standard deviation of $? = 8$ A. What is the z-score corresponding to $X = 52?$ B. What is the X value corresponding to $z = - 0.50?$ C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores? D. What is the z-score corresponding to a sample mean of $M=42$ for a sample of $n = 4$ scores? E. What is the z-scores corresponding to a sample mean of $M= 42$ for a sample of $n = 6$ scores? 2. True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 3. Interpret the location, direction, and distance (near or far) of the following zscores: $a. -2.00 b. 1.25 c. 3.50 d. -0.34$ 4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with $\mu = 78$ and $\sigma = 12$. Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: $82, 74, 62, 68, 79, 94, 90, 81, 80$. 5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about $$12 (\mu = 42, \sigma = 12)$. You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is $44.50 from tips. Test for a difference between this value and the population mean at the $\alpha = 0.05$ level of significance.

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asked 2020-12-07

Suppose you take independent random samples from populations with means $\displaystyle\mu{1}{\quad\text{and}\quad}\mu{2}$ and standard deviations $\displaystyle\sigma{1}{\quad\text{and}\quad}\sigma{2}$. Furthermore, assume either that (i) both populations have normal distributions, or (ii) the sample sizes (n1 and n2) are large. If X1 and X2 are the random sample means, then how does the quantity
$\displaystyle\frac{{{\left(\overline{{{x}_{{1}}}}-\overline{{{x}_{{2}}}}\right)}-{\left(\mu_{{1}}-\mu_{{2}}\right)}}}{{\sqrt{{\frac{{{\sigma_{{1}}^{{2}}}}}{{{n}_{{1}}}}+\frac{{{\sigma_{{2}}^{{2}}}}}{{{n}_{{2}}}}}}}}$
Give the name of the distribution and any parameters needed to describe it.

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asked 2021-02-27

The manager of the store in the preceding exercise calculated the residual for each point in the scatterplot and made a dotplot of the residuals.
The distribution of residuals is roughly Normal with a mean of $0 and standard deviation of $22.92.
The middle 95% of residuals should be between which two values? Use this information to give an interval of plausible values for the weekly sales revenue if 5 linear feet are allocated to the store's brand of men's grooming products.

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asked 2020-11-01

Identify the appropriate hypothesis test for each of the following research situations using the options: The null hypothesis, The Test Statistics, The Sample Statistic, The Standard Error, and The Alpha Level.
A researcher conducts a cross-sectional developmental study to determine whether there is a significant difference in vocabulary skills between 8-year-old and 10-year-old children. A researcher determines that 8% of the males enrolled in Introductory Psychology have some form of color blindness, compared to only 2% of the females. Is there a significant relationship between color blindness and gender?
A researcher records the daily sugar consumption and the activity level for each of 20 children enrolled in a summer camp program. The researcher would like to determine whether there is a significant relationship between sugar consumption and activity level.
A researcher would like to determine whether a 4-week therapy program produces significant changes in behavior. A group of 25 participants is measured before therapy, at the end of therapy, and again 3 months after therapy.
A researcher would like to determine whether a new program for teaching mathematics is significantly better than the old program. It is suspected that the new program will be very effective for small-group instruction but probably will not work well with large classes. The research study involves comparing four groups of students: a small class taught by the new method, a large class taught by the new method, a small class taught by the old method, and a large class taught by the old method.

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asked 2021-01-27

$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}&{H}{o}{u}{s}{e}{w}{\quad\text{or}\quad}{k}{H}{o}{u}{r}{s}\backslash{h}{l}\in{e}{G}{e}{n}{d}{e}{r}&{S}{a}\mp\le\ {S}{i}{z}{e}&{M}{e}{a}{n}&{S}{\tan{{d}}}{a}{r}{d}\ {D}{e}{v}{i}{a}{t}{i}{o}{n}\backslash{h}{l}\in{e}{W}{o}{m}{e}{n}&{473473}&{33.133}{.1}&{14.214}{.2}\backslash{h}{l}\in{e}{M}{e}{n}&{488488}&{18.618}{.6}&{15.715}{.7}\backslash{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$ a. Based on this study, calculate how many more hours per week, on the average, women spend on housework than men. b. Find the standard error for comparing the means. What factor causes the standard error to be small compared to the sample standard deviations for the two groups? The cause the standard error to be small compared to the sample standard deviations for the two groups. c. Calculate the 95% confidence interval comparing the population means for women Interpret the result including the relevance of 0 being within the interval or not. The 95% confidence interval for $\displaystyle{\left(\mu_{{W}}-\mu_{{M}}\right)}$ is: (Round to two decimal places as needed.) The values in the 95% confidence interval are less than 0, are greater than 0, include 0, which implies that the population mean for women could be the same as is less than is greater than the population mean for men. d. State the assumptions upon which the interval in part c is based. Upon which assumptions below is the interval based? Select all that apply. A.The standard deviations of the two populations are approximately equal. B.The population distribution for each group is approximately normal. C.The samples from the two groups are independent. D.The samples from the two groups are random.

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asked 2020-11-20

An automatic machine in a manufacturing process is operating properly if the lengths of an important subcomponent are normally distributed with a mean of 116 cm and a standard deviation of 4.8 cm.
A. Find the probability that one selected subcomponent is longer than 118 cm.
B. Find the probability that if 4 subcomponents are randomly selected, their mean length exceeds 118 cm.
C. Find the probability that if 4 are randomly selected, all 4 have lengths that exceed 118 cm.

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Answer 1

Step 1
Suppose, X be the marks of statistics students, X follows Normal distribution with mean 72 and standard deviation 9.
The average marks of 10 students,
$\displaystyle\overline{{X}}=\frac{{1}}{{10}}{\sum_{{{i}={1}}}^{{10}}}{X}_{{i}}$,
$\displaystyle\overline{{X}}$ follows Normal distributions with mean 72 and standard deviation $\displaystyle\frac{{9}}{{\sqrt{{10}}}}$.
Step 2
The probability that the sample mean between 71 and 73 is
$\displaystyle{P}{\left({71}\le\overline{{X}}\le{73}\right)}$
$\displaystyle={P}{\left(\frac{{{71}-{72}}}{{9}}/{\left(\sqrt{{{10}}}\right)}\le{\left(\frac{{\overline{{X}}-{72}}}{{9}}/{\left(\sqrt{{{10}}}\right)}\le{\left(\frac{{{73}-{72}}}{{9}}/{\left(\sqrt{{{10}}}\right)}\right)},{Z}={\left(\frac{{\overline{{X}}-{72}}}{{9}}/{\left(\sqrt{{{10}}}\right)}\sim{N}{\quad\text{or}\quad}{m}{a}{l}{\left({0},{1}\right)}\right.}\right.}\right.}$
$\displaystyle={P}{\left(-{0.3514}\le{Z}\le{0.3514}\right)}$
$\displaystyle={P}{\left({Z}\le{0.3514}\right)}-{P}{\left({Z}\le-{0.3514}\right)}$
$\displaystyle=\Phi{\left({0.3514}\right)}-\Phi{\left({0.3514}\right)},\Phi{\left({z}\right)}={P}{\left({Z}\le{z}\right)}$
$\displaystyle={2}\Phi{\left({0.3514}\right)}-{1},\Phi{\left({z}\right)}={1}-\Phi{\left(-{z}\right)}$
$\displaystyle={\left({2}\times{0.6374}\right)}-{1}$
=0.2748
The values of $\displaystyle\Phi{\left({0.3514}\right)}$ is taken from Normal distribution.
Therefore, the probability that a random sample of 10 students from this group have a sample mean between 71 and 73 is 0.2758.

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Consider the marks of all 1st-year students on a statistics test. If the marks have a normal distribution with a mean of 72 and a standard deviation of 9, then the probability that a random sample of 10 students from this group have a sample mean between 71 and 73 is?

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