6. A large population (consisting of measurements of diameters of ball bearings) has mean µand standard deviation o, neither of which is perfectly known. A random sample of sizen = 25 observations will be taken, namely the diameters U1, U2, ...,U2, will be observed.In other words, U1, U2, · ,U25, are independent and identically distributed random variableswith u = E(U1) and Var(U1) = o². No further knowledge about the shape of the distributionis known. Define,25T25• (i) If someone says E(X25) = 13, what information does he give you about µ, if any?(ii) If someone says o = 5, what is Var(X25)?(iii) What is the chance, approximately, that a sample of size n = 25 will have its mean,X25, smaller than the population mean µ? In other words, find an approximation ofP(X25 < H). [Hint: see if the value of o is relevant or not to answer the question.](iv) In part (iii) while approximating what theorem did you use, if any?(v) What is the chance, approximately, that a sample of size n = 25 will have its mean,X25, smaller than µ + 2, when o = 5.(vi) From a sample of size n = 25 find an approximate value of P(µ – 2< X, <µ+2),when σ- 10.

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A First Course in Probability (10th Edition)

10th Edition

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Chapter1: Combinatorial Analysis

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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Listed in the accompanying table are weights (kg) of randomly selected U.S. Army male personnel measured in 1988 (from "ANSUR I 1988") and different weights (kg) of randomly selected U.S. Army male personnel measured in 2012 (from "ANSUR II 2012"). Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal. Complete parts (a) and (b). Click the icon to view the ANSUR data. a. Use a 0.05 significance level to test the claim that the mean weight of the 1988 population is less than the mean weight of the 2012 population. What are the null and alternative hypotheses? Assume that population 1 consists of the 1988 weights and population 2 consists of the 2012 weights. A. Ho: H1 H2 H₁₁ H₂ C. Ho H₁ =H2 H₁: H1 H2 OB. Ho: H1 H2 H₁₁₂ OD. Ho: H1 H2 ANSUR data The test statistic is . (Round to two decimal places as needed.) ANSUR II 2012 ANSUR I 1988 80.9 85.6 83.5 73.9 103.7 78.6…
Assume that the readings at freezing on a bundle of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested.If 4.1% of the thermometers are rejected because they have readings that are too high and another 4.1% are rejected because they have readings that are too low, find the two readings that are cutoff values separating the rejected thermometers from the others.interval of acceptable thermometer readings = _________ °C
Please note the question is: What is the distribution of the t-test statistic if m, n -----> ∞ ?
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Assume that the readings at freezing on a bundle of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested.If 4.8% of the thermometers are rejected because they have readings that are too high and another 4.8% are rejected because they have readings that are too low, find the two readings that are cutoff values separating the rejected thermometers from the others.interval of acceptable thermometer readings =_____________ °C
11. Consider a random sample Y₁, Y2, ..., Yn from a normal population Y~N(μ, o²) where the population variance and mean are unknown. We want to construct a Σ(X-X)² Show 100(1 a)% confidence interval for the population variance if g² whether or not is a pivotal quantity and construct a 100(1-a) confidence interval.
The weight (in ounces) of a frozen food package produced by a given manufacturer is a normal random variable with mean u and variance o? (both unknown). We weigh n = 10 of these packages, selected at random and independently from those produced by this manufacturer and find 10 10 E*; = 159, E 4 = 2531. i) Estimate the true average weight of all packages produced by this manufacturer using a 95% confidence interval. ii) Find a 95% upper limit confidence interval for u. iii) Determine a 95% confidence interval for the true standard devi- ation, o.
The random variable, x is a water quality parameter, suspended solids with units of parts per million (ppm). A sample size of 60 yields a mean of 59.87 ppm, and a standard deviation of 12.5 ppm. Water quality criteria requires that the suspended solids do not exceed a mean, µ, = 50 ppm . Test the hypotheses H.: u = µ, vs. HA: u > H. (right-tailed test) at a level of significance of 5 % and assume the pop variance is known.
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3. A machine dispenses a liquid drug into bottles in such a way that the standard deviation of the contents is 45 milliliters. A new machine is tested on a sample of 25 containers and the standard deviation for this sample group is found to be 27 milliliters. At the 2% level of significance, test the claim that the amounts dispensed by the new machine have a smaller standard deviation. (Use χ² - distribution for analysis).
The university data center has two main computers. The center wants to examine whether computer 1 is receiving tasks which require comparable processing time to those of computer 2. A random sample of 11 processing times from computer 1 showed a mean of 35 seconds with a standard deviation of 20 seconds, while a random sample of 7 processing times from computer 2 (chosen independently of those for computer 1) showed a mean of 58 seconds with a standard deviation of 18 seconds. Assume that the populations of processing times are normally distributed for each of the two computers, and that the variances are equal. Can we conclude, at the 0.1 level of significance, that the mean processing time of computer 1, μ1, differs from the mean processing time of computer 2, μ2?Perform a two-tailed test. 1.The null hypothesis: 2.The alternative hypothesis: 3.The value of the test statistic:(Round to at least three decimal places.) 4.The p-value:(Round to at least three decimal places.)

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