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The mean squared error of an estimator
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Chapter 6 Solutions
Probability and Statistics for Engineering and the Sciences
- The teachers at a primary school have devised a particular method to teach arithmetic. They want to find out if this method produces dispersion in mean scores. In order to do so, at first, they took a sample of 25 students who were taught that particular method of arithmetic. Then their test scores were taken into consideration, assuming that the test scores of all the students in that school are approximately normally distributed. Their test score produced an average variance of 180 while the population is assumed to produce a test score of 165. Using a 95% confidence level, can you conclude that the variance of all the test scores of that primary school is different from 165?arrow_forwardMany properties of the fabric used in different textiles, such as sheets and clothing, are influenced by whether the diameter of yarn used in the creation of the textile is consistent. Therefore, various methods can be used to measure the diameter of yarn at specified intervals, such as every 2mm2mm, to determine the consistency of the diameter. These measurements are normally distributed. Suppose that one textile manufacturer will not use any yarn in which the variance of the diameters is greater than 0.0009mm. In order to ensure that the yarn is usable, the diameter of a length of yarn is measured at 100 random intervals. The variance of those measurements is found to be 0.001119mm. Does this evidence provide support that the batch of yarn is unusable by the manufacturer? Use a 0.025 level of significance. Step 3 of 3 : Draw a conclusion and interpret the decision. 1. We reject the null hypothesis and conclude that there is sufficient evidence at a 0.025 level of…arrow_forwardAccording to the records of the last 15 years, the average rainfall in the Marmara region in March is 13 cm and the standard deviation is 4 cm. In the Aegean region, according to the records of the last 10 years, the average precipitation for the same month is 19 cm and the standard deviation is 2 cm. Assuming that the observations were taken from normally distributed masses with different variances, test the hypothesis that the mean precipitation in March is equal with an error of 0.05.arrow_forward
- The number of ice creams sold per hour from Mr Fishy’s van is observed to be a Poisson random variable with parameter λ = 8. Each ice cream costs two pounds but Mr Fishy has to pay five pounds per hour for the pitch. What is the variance of his hourly profit (measured in pounds and ignoring all other income and expenses)?arrow_forwardA researcher would like to conduct a hypothesis test to determine if the mean age of faculty cars is less than the mean age of student cars. A random sample of 25 student cars had a sample mean age of 7 years with a sample variance of 20, and a random sample of 32 faculty cars had a sample mean age of 5.8 years with a sample variances of 16. What is the critical value of the rejection region if the difference is taken as student - faculty and the test is conducted using a 5% significance level?arrow_forwardShow that the mean of a random sample of size n from an exponential population is a minimum variance unbi-ased estimator of the parameter θ.arrow_forward
- 1.Random sample of size 36 were drawn with replacement from a popolation with mean and variance 0.9, and in each case the sample mean was calculated. Find the variance of X. 2.Random sample of size 9 were drawn with replacement from a popolation with mean and variance 4.2, and in each case the sample mean was calculated. Find the variance of X. 3. Random samples of size 10 are drawn with replacement from a population with mean 4 and variance 6. Let X be the sample mean. Find E(X) and Variance(X).arrow_forwardFrom a sample of 25 observations, the estimate of the variance of the population was found to be 15.0 From another sample of 14 observations, the estimate variance was found to be 9.7 Can we accept the hypothesis that the two samples come from populations with equal variance, or must we conclude that the variance of the second population is smaller? Use the 0.01 level of significance.arrow_forwardA sample of 20 elements selected from a population produced a mean of 65 and a standard deviation of 9. Another sample of 25 elements selected from another population produced a mean of 58 and a standard deviation of 11. Assume that the two populations are normally distributed and the standard deviations of the two populations are equal. The null hypothesis is that the means of the two populations are equal and the alternative hypothesis is that the means of the two populations are not equal. The significance level is 5%. The critical values of t are: a. –2.014 and 2.014 b. –2.017 and 2.017 c. –1.681 and 1.681 d. –1.679 and 1.679arrow_forward
- the value used in the equation for the t-test for independent means that approximates the sample size is known as thearrow_forwardThe standard deviation and variance vary in a related way since one can be calculated from the other. The (interquartile range/ standard deviation/ range) is always greater than the (variance/ range/ interquartile range), since the values it uses are always farther apart.arrow_forwardIn a psychological testing experiment, 25 subjects are selected randomly and their reaction time, in seconds, to a particular stimulus is measured. Past experience suggests that the variance in reaction times to these types of stimuli is 4 sec2 and that the distribution of reaction times is approximately normal. The average time for the subjects is 6.2 seconds. Give an upper 95% bound for the mean reaction time.arrow_forward
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