Concept explainers
A shipping company handles containers in three different sizes: (1) 27 ft3 (3 × 3 × 3), (2) 125 ft3, and (3) 512 ft3. Let Xi (i = 1, 2, 3) denote the number of type i containers shipped during a given week. With µi = E(Xi) and σi2 = V(Xi), suppose that the mean values and standard deviations are as follows:
- a. Assuming that X1, X2, X3 are independent, calculate the
expected value and variance of the total volume shipped. [Hint: Volume = 27X1 + 125X2 + 512X3.] - b. Would your calculations necessarily be correct if the Xi’s were not independent? Explain.
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Chapter 5 Solutions
Probability and Statistics for Engineering and the Sciences
- Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data are at or below for respiratory rates in breath per minute during the first 3 years of infancy are given by y=101.82411-0.0125995x+0.00013401x2 for awake infants and y=101.72858-0.0139928x+0.00017646x2 for sleeping infants, where x is the age in months. Source: Pediatrics. a. What is the domain for each function? b. For each respiratory rate, is the rate decreasing or increasing over the first 3 years of life? Hint: Is the graph of the quadratic in the exponent opening upward or downward? Where is the vertex? c. Verify your answer to part b using a graphing calculator. d. For a 1- year-old infant in the 95 th percentile, how much higher is the walking respiratory rate then the sleeping respiratory rate? e. f.arrow_forwardSuppose u, and µ, are true mean stopping distances at 50 mph for cars of a certain type equipped with two different types of braking systems. The data follows: m = 7, x = 113.3, s, = 5.02, n = 7, y = 129.8, and s, = 5.32. Calculate a 95% CI for the difference between true average stopping distances for cars equipped with system 1 and cars equipped with system 2. (Round your answers to two decimal places.) n USE SALT |-22.52 |-10.48 Does the interval suggest that precise information about the value of this difference is available? O Because the interval is so wide, it appears that precise information is available. O Because the interval is so narrow, it appears that precise information is not available. O Because the interval is so wide, it appears that precise information is not available. O Because the interval is so narrow, it appears that precise information is available.arrow_forwardA photoconductor film is manufactured at a nominal thickness of 25 mils. The product engineer wishes to increase the mean speed of the film and believes that this can be achieved by reducing the thickness of the film to 20 mils. Eight samples of each film thickness are manufactured in a pilot production process, and the film speed (in microjoules per square inch) is measured. For the 25-mil film, the sample data result is.x₁ = 1.15 and S₁ = 0.11, while for the 20-mil film, the data yield 2 = 1.06 and s2 = 0.09. Note that an increase in film speed would lower the value of the observation in microjoules per square inch. Do the data support the claim that reducing the film thickness increases the mean speed of the film? Use a = 0.10 and assume that the two population variances are equal and the underlying population of film speed is normally distributed. The appropriate decision for the test is to reject the null hypothesis True Falsearrow_forward
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- Total plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that a random sample of 46 male firefighters are tested and that they have a plasma volume sample mean of x = 37.5 ml/kg (milliliters plasma per kilogram body weight). Assume that σ = 7.00 ml/kg for the distribution of blood plasma. (a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Round your answers to two decimal places.) lower limit upper limit margin of error (b) What conditions are necessary for your calculations? (Select all that apply.) the distribution of weights is normalthe distribution of weights is uniformσ is unknownn is largeσ is known (c) Interpret your results in the context of this problem. We are 1% confident that the true…arrow_forwardTotal plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that a random sample of 48 male firefighters are tested and that they have a plasma volume sample mean of x = 37.5 ml/kg (milliliters plasma per kilogram body weight). Assume that σ = 7.10 ml/kg for the distribution of blood plasma. (a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Round your answers to two decimal places.) lower limit upper limit margin of error (b) What conditions are necessary for your calculations? (Select all that apply.) n is largethe distribution of weights is uniformσ is knownthe distribution of weights is normalσ is unknown (c) Interpret your results in the context of this problem. We are 99% confident that the true…arrow_forwardQ1 (а) і. 10,000 vehicle speed traffic data were collected during the weekday on route FT050 at Batu Pahat, Johor. Suggest how to quantify and describe the basic characteristic of this data. Then explain how the method you suggested can help others to understand the data.arrow_forward
- Total plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that a random sample of 42 male firefighters are tested and that they have a plasma volume sample mean of x = 37.5 ml/kg (milliliters plasma per kilogram body weight). Assume that σ = 8.00 ml/kg for the distribution of blood plasma. (a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Round your answers to two decimal places.) lower limit upper limit margin of error (b) What conditions are necessary for your calculations? (Select all that apply.) σ is knownthe distribution of weights is normalσ is unknownn is largethe distribution of weights is uniform (c)Find the sample size necessary for a 99% confidence level with maximal margin of error E =…arrow_forwardOcean currents are important in studies of climate change, as well as ecology studies of dispersal of plankton. Drift bottles are used to study ocean currents in the Pacific near Hawaii, the Solomon Islands, New Guinea, and other islands. Let x represent the number of days to recovery of a drift bottle after release and y represent the distance from point of release to point of recovery in km/100. The following data are representative of one study using drift bottles to study ocean currents. Σx = 476, Σy = 87.1, Σx2 = 62,290, Σy2 = 2046.87, Σxy = 11121.3,and r ≈ 0.94367. x days 72 76 32 91 205y km/100 14.7 19.6 5.3 11.7 35.8 a) Use a 1% level of significance to test the claim ? > 0.(Use 2 decimal places.)t =critical t= b) Find the predicted distance (km/100) when a drift bottle has been floating for 60 days. (Use 2 decimal places.)________ km/100 c) Find a 90% confidence interval for your prediction of part (d). (Use 1 decimal place.)lower limit = _____…arrow_forwardTotal plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that a random sample of 44 male firefighters are tested and that they have a plasma volume sample mean of x = 37.5 ml/kg (milliliters plasma per kilogram body weight). Assume that σ = 7.70 ml/kg for the distribution of blood plasma. (a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Round your answers to two decimal places.) lower limit upper limit margin of error (b) What conditions are necessary for your calculations? (Select all that apply.) the distribution of weights is normal σ is known the distribution of weights is uniform n is large σ is unknown (c) Find the sample size necessary for a 99% confidence level with maximal margin of…arrow_forward
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