Nishank V.

Mathematics Major

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Pre-Calculus

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Question:

If the value of b*h = 2400, and P = 2b + 3h. Find the minimum value of P.

Nishank V.

Answer:

P = 2b + 3h First we convert the equation into only one variable. P = 2b + 3(2400 / b) P = 2b + 3*2400b⁻¹ P = 2b + 7200b⁻¹ To find the minimum, dP/db = 0 dP/db = 2 - 7200b⁻² = 0 So, 7200 b⁻² = 2 b = √3600 b = 60 h = 2400/b = 2400/60 = 40 So, b = 60, h = 40 P = 120 + 120 = 240

Statistics

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Question:

The percent of a population having a given medical condition is 1.8%. The accuracy of diagnostic test for the condition is such that • The diagnostic test will yield a “positive” test result in 99.5% of cases where the individual being tested actually has the medical condition. • The diagnostic test will yield a “positive” test result in only 0.16% of cases where the individual being tested does not have the medical condition. Given that an individual tests positive, what is the probability that they do not actually have the medical condition?

Nishank V.

Answer:

We use Bayes' theorem, which helps us find the probability of event A given event B [written P(A|B)], in terms of the probability of B given A [written P(B|A)], and the probabilities of A & B P(A|B) = [P(A)P(B|A)] / P(B) Here, A = No condition, B = Tested positive P(No condition | Positive) = [P(No condition)*P(Positive | No condition)] / P(Positive) Now, P(No condition) = 100-1.8% = 98.2% = 0.982 P(Positive | No condition) = 0.16% = 0.0016 P(Positive) = P(Positive | No condition)*P(No condition) + P(Positive | have condition)*P(Condition) = 0.0016*0.982 + 0.995*0.018 = 0.0195 So, P(No condition | Positive) = [0.982*0.0016]/0.0195 = 0.0806 = 8.06%

Statistics

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Question:

The distributions of systolic and diastolic blood pressure for female diabetics between the ages of 30 and 34 have unknown means. However, the sample means for each of the systolic and diastolic blood pressure are 130 and 84 mmHg and their standard deviations are known to be 11.8 and 9.1 mmHg, respectively. A random sample of size 10 is selected from this population. Complete the following - a) Calculate a 95% CI on the mean diastolic blood pressure. Interpret this interval. b) Calculate a 99% CI on the mean systolic blood pressure. Interpret this interval. c) Calculate a 90% CI on the mean systolic blood pressure. Interpret this interval. d) Compare parts b) and c). e) What assumptions are necessary to compute the confidence intervals in part a)?

Nishank V.

Answer:

a) Calculate a 95% CI on the mean diastolic blood pressure. Interpret this interval. The t value for 95% confidence with df = 9 is t = 2.262 CI = 84 ± 2.262*(9.1/√10) = 84 ± 6.51 = (77.49, 90.51) Interpretation : Based on this sample of size n=10, our best estimate of the true mean diastolic blood pressure in the population is 84. Based on this sample, we are 95% confident that the true systolic blood pressure in the population is between 77.49 and 90.51 b) Calculate a 99% CI on the mean systolic blood pressure. Interpret this interval. The t value for 99% confidence with df = 9 is t = 3.250 CI = 130 ± 3.250*(11.8/√10) = 130 ± 12.13 = (117.87, 142.13) Interpretation: Based on this sample of size n=10, our best estimate of the true mean systolic blood pressure in the population is 130. Based on this sample, we are 99% confident that the true systolic blood pressure in the population is between 117.87 and 142.13 c) Calculate a 90% CI on the mean systolic blood pressure. Interpret this interval. The t value for 90% confidence with df = 9 is t = 1.833 CI = 130 ± 1.833*(11.8/√10) = 130 ± 6.84 = (123.16, 136.84) Interpretation : Based on this sample of size n=10, our best estimate of the true mean systolic blood pressure in the population is 130. Based on this sample, we are 90% confident that the true systolic blood pressure in the population is between 123.16 and 136.84 d) Compare parts b) and c). The confidence interval for 99% is larger than the confidence interval for 90%, as expected e) What assumptions are necessary to compute the confidence intervals in part a)? The sample size is small, i.e. n < 30

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