A company packages powdered soap in 6-pound boxes. The sample mean and standard

If \(\bar{X}\) and \(\bar{Y}\) are the respective means of two independent random samples of the same size \(n\), find \(n\) if we want \(\bar{x}-\bar{y} \pm 4\) to be a \(90 \%\) confidence interval for \(\mu_{X}-\mu_{Y}\). Assume that the standard deviations are known to be \(\sigma_{X}=15\) and \(\sigma_{Y}=25\). Equation Transcription: Text Transcription: Bar X bar Y n Bar x-bar y pm 4 90% mu_X-mu_Y sigma_ X=15 sigma_Y=25

PROBLEM 2E Let X equal the excess weight of soap in a “1000- gram” bottle. Assume that the distribution of X is N(?, 169). What sample size is required so that we have 95% confidence that the maximum error of the estimate of ? is 1.5?

PROBLEM 3E A company packages powdered soap in “6-pound” boxes. The sample mean and standard deviation of the soap in these boxes are currently 6.09 pounds and 0.02 pound, respectively. If the mean fill can be lowered by 0.01 pound, $14,000 would be saved per year. Adjustments were made in the filling equipment, but it can be assumed that the standard deviation remains unchanged. (a) How large a sample is needed so that the maximum error of the estimate of the new ? is ? = 0.001 with 90% confidence? (b) A random sample of size n = 1219 yielded x = 6.048 and s = 0.022. Calculate a 90% confidence interval for ?. (c) Estimate the savings per year with these new adjustments. (d) Estimate the proportion of boxes that will now weigh less than 6 pounds.

PROBLEM 7E For a public opinion poll for a close presidential election, let p denote the proportion of voters who favor candidate A. How large a sample should be taken if we want the maximum error of the estimate of p to be equal to (a) 0.03 with 95% confidence? (b) 0.02 with 95% confidence? (c) 0.03 with 90% confidence?

Measurements of the length in centimeters of \(n=29\) fish yielded an average length of \(\bar{x}=16.82\0 and \(s^{2}=\) 34.9. Determine the size of a new sample so that \(\bar{x} \pm 0.5\) is an approximate \(95 \%\) confidence interval for \(\mu\). Equation Transcription: Text Transcription: n=29 barx=16.82 s^2=34.9 Bar x pm 0.5 95% mu

A quality engineer wanted to be \(98 \%\) confident that the maximum error of the estimate of the mean strength, \(\mu\), of the left hinge on a vanity cover molded by a machine is \(0.25\). A preliminary sample of size \(n=32\) parts yielded a sample mean of \(\bar{x}=35.68\) and a standard deviation of \(s=1.720\). (a) How large a sample is required? (b) Does this seem to be a reasonable sample size? (Note that destructive testing is needed to obtain the data.) Equation Transcription: Text Transcription: 98% Mu 0.25 n=32 Bar x=35.68 s=1.723

A manufacturer sells a light bulb that has a mean life of 1450 hours with a standard deviation of \(33.7\) hours. A new manufacturing process is being tested, and there is interest in knowing the mean life \(\mu\) of the new bulbs. How large a sample is required so that \(\bar{x} \pm 5\) is a \(95 \%\) confidence interval for \(\mu\)? You may assume that the change in the standard deviation is minimal. Equation Transcription: Text Transcription: 33.7 Bar x pm 5 95% mu

PROBLEM 1E Let X equal the tarsus length for a male grackle. Assume that the distribution of X is N(?, 4.84). Find the sample size n that is needed so that we are 95% confident that the maximum error of the estimate of ? is 0.4.

PROBLEM 8E Some college professors and students examined 137 Canadian geese for patent schistosome in the year they hatched. Of these 137 birds, 54 were infected. The professors and students were interested in estimating p, the proportion of infected birds of this type. For future studies, determine the sample size n so that the estimate of p is within ? = 0.04 of the unknown p with 90% confidence.

PROBLEM 9E A die has been loaded to change the probability of rolling a 6. In order to estimate p, the new probability of rolling a 6, how many times must the die be rolled so that we are 99% confident that the maximum error of the estimate of p is ? = 0.02?

If \(Y_{1} / n\) and \(Y_{2} / n\) are the respective independent relative frequencies of success associated with the two binomial distributions \(b\left(n, p_{1}\right)\) and \(b\left(n, p_{2}\right)\), compute \(n\) such that the approximate probability that the random interval \(\left(Y_{1} / n-Y_{2} / n\right) \pm 0.05\) covers \(p_{1}-p_{2}\) is at least \(0.80\). Hint: Take \(p_{1}^{*}=p_{2}^{*}=1 / 2\) to provide an upper bound for \(n\). Equation Transcription: Text Transcription: Y_1/n Y_2/n b(n,p_1) b(n,p_2) (Y_1/n-Y_2/n)pm0.05 p_1-p_2 0.80 p-1^*=p_2^*=1/2 n

A seed distributor claims that 80% of its beet seeds will germinate. How many seeds must be tested for germination in order to estimate p, the true proportion that will germinate, so that the maximum error of the estimate is \(\varepsilon = 0.03\) with 90% confidence?

Some dentists were interested in studying the fusion of embryonic rat palates by a standard transplantation technique. When no treatment is used, the probability of fusion equals approximately \(0.89\). The dentists would like to estimate \(p\), the probability of fusion, when vitamin \(\mathrm{A}\) is lacking. (a) How large a sample \(n\) of rat embryos is needed for \(y / n \pm 0.10\) to be a \(95 \%\) confidence interval for \(p\) ? (b) If \(y=44 \) out of \(n=60\) palates showed fusion, give a \(95 \%\) confidence interval for \(p\). Equation Transcription: Text Transcription: 0.89 p y/n pm 0.10 95% y=44 n=60

PROBLEM 12E Let p equal the proportion of college students who favor a new policy for alcohol consumption on campus. How large a sample is required to estimate p so that the maximum error of the estimate of p is 0.04 with 95% confidence when the size of the student body is (a) N = 1500? (b) N = 15,000? (c) N = 25,000?

PROBLEM 13E Out of 1000 welds that have been made on a tower, it is suspected that 15% are defective. To estimate p, the proportion of defective welds, how many welds must be inspected to have approximately 95% confidence that the maximum error of the estimate of p is 0.04?

Let X equal the tarsus length for a male grackle. Assume that the distribution of X is N(, 4.84). Find the sample size n that is needed so that we are 95% confident that the maximum error of the estimate of is 0.4.

Let X equal the excess weight of soap in a 1000- gram bottle. Assume that the distribution of X is N(, 169). What sample size is required so that we have 95% confidence that the maximum error of the estimate of is 1.5?

A company packages powdered soap in 6-pound boxes. The sample mean and standard deviation of the soap in these boxes are currently 6.09 pounds and 0.02 pound, respectively. If the mean fill can be lowered by 0.01 pound, $14,000 would be saved per year. Adjustments were made in the filling equipment, but it can be assumed that the standard deviation remains unchanged. (a) How large a sample is needed so that the maximum error of the estimate of the new is = 0.001 with 90% confidence? (b) A random sample of size n = 1219 yielded x = 6.048 and s = 0.022. Calculate a 90% confidence interval for . (c) Estimate the savings per year with these new adjustments. (d) Estimate the proportion of boxes that will now weigh less than 6 pounds.

Measurements of the length in centimeters of n = 29 fish yielded an average length of x = 16.82 and s2 = 34.9. Determine the size of a new sample so that x 0.5 is an approximate 95% confidence interval for

A quality engineer wanted to be 98% confident that the maximum error of the estimate of the mean strength, , of the left hinge on a vanity cover molded by a machine is 0.25. A preliminary sample of size n = 32 parts yielded a sample mean of x = 35.68 and a standard deviation of s = 1.723. (a) How large a sample is required? (b) Does this seem to be a reasonable sample size? (Note that destructive testing is needed to obtain the data.)

A manufacturer sells a light bulb that has a mean life of 1450 hours with a standard deviation of 33.7 hours. A new manufacturing process is being tested, and there is interest in knowing the mean life of the new bulbs. How large a sample is required so that x5 is a 95% confidence interval for ? You may assume that the change in the standard deviation is minimal.

For a public opinion poll for a close presidential election, let p denote the proportion of voters who favor candidate A. How large a sample should be taken if we want the maximum error of the estimate of p to be equal to (a) 0.03 with 95% confidence? (b) 0.02 with 95% confidence? (c) 0.03 with 90% confidence?

Some college professors and students examined 137 Canadian geese for patent schistosome in the year they hatched. Of these 137 birds, 54 were infected. The professors and students were interested in estimating p, the proportion of infected birds of this type. For future studies, determine the sample size n so that the estimate of p is within \(\varepsilon = 0.04\) of the unknown p with 90% confidence.

A die has been loaded to change the probability of rolling a 6. In order to estimate p, the new probability of rolling a 6, how many times must the die be rolled so that we are 99% confident that the maximum error of the estimate of p is \(\varepsilon = 0.02\)?

A seed distributor claims that 80% of its beet seeds will germinate. How many seeds must be tested for germination in order to estimate p, the true proportion that will germinate, so that the maximum error of the estimate is = 0.03 with 90% confidence?

Some dentists were interested in studying the fusion of embryonic rat palates by a standard transplantation technique. When no treatment is used, the probability of fusion equals approximately 0.89. The dentists would like to estimate p, the probability of fusion, when vitamin A is lacking. (a) How large a sample n of rat embryos is needed for y/n 0.10 to be a 95% confidence interval for p?(b) If y = 44 out of n = 60 palates showed fusion, give a 95% confidence interval for p.

Let p equal the proportion of college students who favor a new policy for alcohol consumption on campus. How large a sample is required to estimate p so that the maximum error of the estimate of p is 0.04 with 95% confidence when the size of the student body is (a) N = 1500? (b) N = 15,000? (c) N = 25,000?

Out of 1000 welds that have been made on a tower, it is suspected that 15% are defective. To estimate p, the proportion of defective welds, how many welds must be inspected to have approximately 95% confidence that the maximum error of the estimate of p is 0.04?

If Y1/n and Y2/n are the respective independent relative frequencies of success associated with the two binomial distributions b(n, p1) and b(n, p2), compute n such that the approximate probability that the random interval (Y1/n Y2/n) 0.05 covers p1 p2 is at least 0.80. Hint: Take p 1 = p 2 = 1/2 to provide an upper bound for n.

If X and Y are the respective means of two independent random samples of the same size n, find n if we want \(\bar x - \bar y \pm 4\) to be a 90% confidence interval for \(\mu_X - \mu_Y\). Assume that the standard deviations are known to be \(\sigma_X = 15\) and \(\sigma_Y = 25\).