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Math 117 Sample Exam Three PAGE ONE
In Problems (1)-(13) please choose as many options as appropriate, from amongst the following probability distributions: (A) Binomial (B) Hypergeometric (C) Continuous Uniform (D) Normal (E) t (F) CHI – Square (G) DISCRETE UNIFORM
(1) Which discrete distribution involves sampling WITHOUT replacement ? ?
(2) Which distributions REQUIRE a table to help find probabilities of events?
(3) Which distributions are ALWAYS symmetric?
(4) Which discrete distribution is associated , by ITS NAME, with EXACTLY TWO possible outcomes on each trial?
(5) Which distributions are, IN GENERAL, NOT symmetric?
(6) Which DISCRETE distribution is associated with DEPENDENT trials ?
(7) Which distributions are CONTINUOUS?
(8) Which continuous distribution ALLOWS us to determine probabilities of events by merely computing areas of rectangles?
(9) What is the probability distribution of the SAMPLE MEAN, if: UNDERLYING POPULATION IS NORMAL, POP. VARIANCE IS KNOWN and sample size=23.
(10) Which discrete distribution is associated with INDEPENDENT trials??
(11) Which continuous distributions have a curve, which varies as the sample size varies?
(12) What is the probability distribution of the SAMPLE MEAN, if: UNDERLYING POPULATION IS NORMAL, POP. VARIANCE IS NOT KNOWN, and sample size=15.
(13) Which DISCRETE distribution involves EQUALLY LIKELY outcomes ?
(14) The standard normal curve Z has WHAT numerical value for its :
A. MEAN? B. VARIANCE?
(15) Suppose that X is normally distributed, with MEAN=μ and VARIANCE=σ?2. Please describe, IN WORDS, how one ARITHMETICALLY converts some given “x” DATA VALUE of the random variable X into a “z” value for the random variable Z. PLEASE BE VERY CLEAR AND VERY ARITHMETICALLY PRECISE IN YOUR ARITHMETIC DESCRIPTIONS.
(16) Suppose that X is BINOMIALLY DISTRIBUTED. What value must the binomial parameter representing PROB (a success occurs on a given trial) have, in order that the random variable X be symmetric?
(17) As the standard deviation of a normal random variable X DECREASES, the SHAPE of the corresponding normal curve:
(i) EXPANDS (ii) CONDENSES (iii) IS UNCHANGED
(18) Suppose that X is normally distributed, with mean 94.6 . PLEASE IDENTIFY THE TWO “TAIL” REGIONS that are contained in the following four scenarios:
(i) P(X<101) (ii) P(X<92.7) (iii) P(X>84.9) (iv) P(X>95.2)
(19) In the binomial distribution , the X value having the highest probability
should be the one that is either equal to or very close to which parameter?
(20) Does the standard deviation of the probability distribution of x?: (i)
(i) DECREASE or (ii) INCREASE , as the sample size decreases?
MATH 117 SAMLE EXAM THREE PAGE TWO
(21) X is normally distributed, with mean 62. Therefore, compare the normal curve’s: (A) Area to the right of 56.5; to the normal curve’s: (B) Area to the left of 64.5;
(i) Area (A) exceeds Area (B) (ii) Area (B) exceeds Area (A)
(iii) Area (A) =Area (B) (iv) We would need to know the value of σ, in order to compare Areas (A) and (B)
(22) A random variable, whose possible outcomes must be MEASURED, but NOT counted, is referred to as what?
(23)–(26) X is a BINOMIAL random variable having: sample size=75; PROB (success in ANY TRIAL)=.8
(23A) Find the average value of X .
(23B) Find the Variance of X .
(24)Find PROB (X is ONE or HIGHER)? Please DO NOT BOTHER to carry out any of the decimal arithmetic. Hint: Please calculate the EASY WAY!
(25) Write the EXACT probability expression for PROB (X is AT LEAST 51,BUT less than 64) using the ∑ symbol. DO NOT ATTEMPT TO DO ANY COMPUTATION
(26)BLUNTLY and CLEARLY EXHIBIT the arithmetic of the THREE inequalities, which authorize us to use the normal approximation to the binomial distribution, for the given binomial random variable.
(27)IF we are APPROPRIATELY using the t distribution to help determine a confidence
interval for a mean of a given population, then the VARIANCE of that population IS;
(i) not known (ii) known (iii) insufficient information to respond to this question
(28)If we INCREASE the sample size, in a given experiment, in order to ultimately
determine a confidence interval for the population mean, then the corresponding
MAXIMUM ERROR, in using the sample mean as an estimate of the population
mean, will be:
(i) increased (ii) lowered (iii) unchanged (iv) insufficient information to respond
(29) Compare and , IN GENERAL.
(i) (ii) (iii) ≥ (iv) insufficient information
(30)Compare and , IN GENERAL.
(i) (ii) (iii) (iv) insufficient information to respond
(31) H is normally distributed, with mean = . IF the “z” value for a corresponding data value h happens to be NEGATIVE, then that data value h must be:
a. HIGHER THAN b. EQUAL TO
c. LOWER THAN d. Relationship between h and is NOT CLEAR
(32) What “z” value do we use, if we wish to have 98% confidence regarding the value of the population mean?
(33) What “z” value do we use, if we wish to have 80% confidence regarding the value of the population mean?
(34) Which confidence interval, the 80% CI or the 98% CI, must have a LARGER maximum error of estimate? Briefly explain your response.
MATH 117 SAMPLE EXAM THREE PAGE THREE
(35) We have just computed the 95% confidence interval for the population mean. The outcome: LOWER CONFIDENCE LIMIT = 64 and UPPER CONFIDENCE LIMIT = 78.
a. What information/usefulness does the above statistical analysis provide us with? THAT IS, WHY would one even BOTHER to make such a calculation?
b. What is the maximum error of estimate, for this analysis?
c. What is the sample average for the given data?
d. IF the population standard deviation = 35, then with the assistance of your (b) and/or (c) responses, please determine the sample size for this analysis.
(36) You wish to locate SOME appropriate pair of entries from the table, in order to obtain a confidence interval for WHICH parameter?
(37) Continuing with the notion in the previous question, your sample size is 26 and you wish 99% confidence. Which two tabular entries are relevant to your CI calculation?
(38) Continuing with the notion in the previous TWO questions, you must either KNOW or MAKE SOME ASSUMPTION about some underlying characteristic of the population from which you wish to sample. What characteristic must our population have, in order that the above CI calculation in (36) /(37) would be deemed reasonable, statistically speaking?
(39) Determine the EXACT measure of AREA, UNDER the curve, ABOVE the horizontal axis, and to the RIGHT of zero.
(40) We have JUST determined the UPPER and LOWER confidence limits for the VARIANCE of some population interest. EXACTLY WHAT DO WE DO to each of the confidence limits, in order that we be able to determine the corresponding confidence limits for the STANDARD DEVIATION of this population ?
(41) For a given set of data, you have just determined the statistics:
(1) SAMPLE MEAN; and (2) MAXIMUM ERROR of ESTIMATE, towards determining a t distribution oriented confidence interval for the POPULATION AVERAGE. The random sample size is 21.
(i)Which TWO numerical calculations must you perform, involving the SAMPLE MEAN and MAXIMUM ERROR, in order to actually determine the appropriate confidence limits for your CI?
(ii) Relative to the value of the MAXIMUM ERROR, how does one determine the length of the confidence interval?
(iii) What is the number of degrees of freedom for this confidence interval analysis?
(42) You wish to determine an 80% confidence interval for a (BINOMIAL) population proportion.
(i) Which set of three inequalities should we check on, in order to justify the CI
calculations as being made with the assistance of an appropriate “z” value?
(ii) CARRY OUT the arithmetic for these three inequalities, assuming a random
sample size of 118 independent trials led to 47 “successful” outcomes.
(iii) What is the SAMPLE proportion of success, for these 118 trials (3 decimals)?
(iv) What is the numerical value of the DIFFERENCE between the SAMPLE
PROPORTION OF SUCCESS and the LOWER CONFIDENCE LIMIT for the 80%CI?
MATH 117 SAMPLE EXAM THREE PAGE FOUR
(43) R is a CONTINUOUS UNIFORM random variable, with possible outcomes between 35 and 95. Please determine:
(i) VARIANCE of outcomes? (ii) AVERAGE outcome? (iii) PROB (outcome R exceeds 32.8)? (iv) PROB (outcome R is NOT between 43.5 and 57.5)?
(44) In the use of the MEGASTAT/EXCEL computer methods, I found a sample size “n”=9, and probability of success “p”, which yield the following data for the associated BINOMIAL probability distribution:
Cumulative Probability
0 0.0404 0.0404
1 0.1557 0.1961
2 0.2668 0.4629
3 0.2668 0.7297
4 0.1715 0.9012
5 ? 0.9747
6 0.0210 0.9957
7 0.0039 ?
8 0.0004 ?
9 0.00002 ?
(i) Between which two values of must the MEAN value of this distribution lie?
(ii) Using ONLY the decimals provided in the chart, determine the probability that = 6?
(iii) Using ONLY the decimals provided in the chart, determine the CUMULATIVE probability related to
(iv) If you were to look at the binomial histogram, would you think that the distribution would be symmetric? That is, do you perceive that Briefly explain your response.
(v) What is prob (5) ? (vi) What is the TOTAL measure of the UNACCOUNTED cumulative probability, to the best 4 decimal places?
(vii) How many independent trials are involved with this binomial experiment ?
(45) In the use of the MEGASTAT/EXCEL computer methods, suppose the computer is set to the use of the STANDARD NORMAL CURVE and requests a Z value from you, while providing you with the associated probability as a response. Suppose you enter a Z value of -0.68 . Therefore, please anticipate the computer’s response:
(i) Please draw the associated standard normal curve picture, while shading the appropriate area under the standard normal curve.
(ii) Please provide me with that measure of shaded area.
(46) In the use of the MEGASTAT/EXCEL computer methods, suppose the computer is set to the analysis of a normal curve X having a mean of 39 and standard deviation = 14. Also, the computer is set to calculate X, given . Thus, you have entered, in the three boxes noted at the top of the next page :
MATH 117 SAMPLE EXAM THREE PAGE FIVE
(46) CONTINUED
0.19
Mean 39
Standard deviation 14
(i) What data value do you anticipate the computer would provide you with?
(ii)What is the associated Z value, for the data value you noted in (i)?
(iii) Please draw the associated standard normal curve picture, while showing the appropriate Z value, and also illustrating the appropriate shaded area under the standard normal curve AND its mean.
(47) In the use of the MEGASTAT/EXCEL computer methods, suppose you indicate to the computer that you wish to determine a confidence interval for the population mean, using a “ ” analysis, with 99% confidence, using a sample size of 12. The computer ultimately provides you with the following statistical analysis :
Half- width = 8.9663 ; lower confidence limit =18.0337 ; upper confidence limit = 35.9663. Please determine each of the following statistics:
(a) Standard deviation of the data ? (b) length of the confidence interval ?
(b) sample average of the data ?
In the ” SAMPLE MEAN FLOWCHART“ problems (48) – (51), please answer the three questions indicated as question marks ( ? ) , in the last 3 columns of each question.
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