Discuss the progression of statistics and probability from ancient times to modern times including a discussion of the uses of statistics and probability prior to the foundations in the 16^th and 17^th centuries.

A Cessna 172 has the following characteristics: Wingspan: 36’ 1” Wing Area: 174 ft2 Normal Gross Weight: 2,450 lb Fuel Capacity: 54 gal of aviation gasoline Power Plant: 1 × Lycoming IO–360–L2A piston engine, 180 hp Specific Fuel Consumption: 0.3 lb/hp-h Parasite drag coefficient: CD,0 = 0.037 Oswald efficiency factor: e = 0.72 Propeller efficiency: 0.8 #Passengers: 3 (+1 pilot)

(a) Produce a plot showing the power required vs. velocity of the aircraft. Label the axes clearly. [Use Excel or a programming language of your choice]

(b) Produce a plot showing the L/D ratio vs. velocity. Label the axes clearly. [Use Excel or a programming language of your choice]

(c) What is the maximum velocity at sea level?

(d) What is the minimum flight velocity at sea level?

(e) What is the maximum velocity at 5,000 ft?

(f) What is the maximum rate of climb?

(g) What is the service ceiling?

(h) How long does it take to get to the service ceiling?

(i) If you run out of fuel at an altitude of 5,000 ft, how far can the plane glide?

(j) What is the maximum range?

(k) What is the range at maximum velocity?

(l) What is the maximum endurance?

(m) What distance is required for takeoff?

(n) What distance is required for landing?

A 0.020 mol sample of carbon monoxide (CO) is placed in a cylinder fitted with a movable frictionless piston. This gaseous sample - initially at volume V_i = 4.00 L - has a pressure of 0.400 atm - identical to the external pressure. This gaseous sample then undergoes a cyclic process. Throughout the cyclic process, the sample is thermostatted at a Kelvin temperature T. The details of this cyclic process:

First, a two-step expansion carried out as follows:   
The external pressure is decreased to 0.200 atm and is held constant at this value while the volume change takes place. The first step involves a volume change from V_i to V₁.

Second:
The external pressure is decreased to 0.100 atm and is held constant at this value while the volume change takes place. The second step involves a volume change from V₁ to V₂ .

Next, a one-step compression is carried out:
The external pressure is increased to a value of P_{ext 1C} and is held constant at this value while the volume change takes place. One-step compression involves a volume change from V₂ to a final volume V_f.  P_{ext 1C} and V_f are consistent with the overall process being a cyclic one.

(a) Determine V₁ (in L), V₂ (in L), Vf (in L), and P_{ext 1C} (in atm) for this system and work for each step in the cycle: w_1E , w_2E , and w_1C .  

(b) Sketch the two-step expansion on P-V graph

c) Sketch the one-step compression on P-V graph
  

1. In an isobaric compression of an idea gas, a) the internal energy of the gas remains constant,    b) no heat flows into the gas, c) no work is done on the gas, d) work is done on the gas,        and/or e) work is done by the gas.

2. An ideal gas undergoes an isothermal expansion from one state to another.  In this process determine the following (using the sign conventions on page 413):
   • Q = 0, Q > 0 or Q < 0
   • W = 0, W > 0 or W < 0
   • ΔU = 0, ΔU > 0 or ΔU < 0

3. An ideal gas undergoes an isothermal process.  Which of the following are true (may be more than one):  a) No heat is added or removed from the gas, b) the internal energy of the gas does not change and/or c) the average kinetic energy of the molecules does not change.

4. An ideal gas undergoes an adiabatic expansion in which no heat flows into or out of the gas.  As a result determine which of the following properties of the gas increases (may be more than one): a) temperature, b) pressure and/or c) volume.  

5. Can you cool your kitchen by leaving your refrigerator door open?  Why or why not?

6. If 1500 J of heat is added to a system and the system does 1100 J of work, then what was the change in the internal energy of the system?   Assume the system is closed (i.e. a gas inside of a cylinder fitted with a movable piston).

Thirty-two small communities in Connecticut (population near 10,000 each) gave an average of x = 138.5 reported cases of larceny per year. Assume that σ is known to be 44.9 cases per year.

(a) Find a 90% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

(b) Find a 95% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

(c) Find a 99% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

(d) Compare the margins of error for parts (a) through (c). As the confidence levels increase, do the margins of error increase?

As the confidence level increases, the margin of error remains the same.As the confidence level increases, the margin of error decreases.     As the confidence level increases, the margin of error increases.

(e) Compare the lengths of the confidence intervals for parts (a) through (c). As the confidence levels increase, do the confidence intervals increase in length?

As the confidence level increases, the confidence interval increases in length.As the confidence level increases, the confidence interval remains the same length.     As the confidence level increases, the confidence interval decreases in length.

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 15 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with σ = 0.22 gram.

When finding an 80% confidence interval, what is the critical value for confidence level? (Give your answer to two decimal places.)

z_c =

(a)

Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.)

lower limitupper limitmargin of error

(b)

What conditions are necessary for your calculations? (Select all that apply.)

normal distribution of weightsuniform distribution of weightsσ is knownσ is unknownn is large

(c)

Interpret your results in the context of this problem.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.There is a 20% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.     There is an 80% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.The probability to the true average weight of Allen's hummingbirds is equal to the sample mean.

(d)

Which equation is used to find the sample size n for estimating μ when σ is known?

n =

n =

     n =

n =

Find the sample size necessary for an 80% confidence level with a maximal margin of error  E = 0.08 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)

hummingbirds

Suppose in a local Kindergarten through 12^th grade (K - 12) school district, 53 percent of the population favor a charter school for grades K through five. A simple random sample of 800 is surveyed. Calculate the following using the normal approximation to the binomial distribution. (Round your answers to four decimal places.)

(a) Find the probability that less than 340 favor a charter school for grades K through 5.

(b) Find the probability that 415 or more favor a charter school for grades K through 5.

(c) Find the probability that no more than 390 favor a charter school for grades K through 5.


(d) Find the probability that there are fewer than 375 that favor a charter school for grades K through 5.


(e) Find the probability that exactly 400 favor a charter school for grades K through 5.

The overhead reach distances of adult females are normally distributed with a mean of 205.5 cm and a standard deviation of 8.3 cm.

A. Find the probability that an individual distance is greater than 218.00 cm.

B. Find the probability that the mean for 20 randomly selected distances is greater than 203.30 cm.

C. Why can the normal distribution be used in part b) even though the sample size does not exceed 30?

a. The probability is?

round to four decimal places

b. The probability is?

round to four decimal places

c. Choose one.

A The normal distribution can be used because the mean is large.

B The normal distribution can be used because the finite population correction factor is small.

C. The normal distribution can be used because the probability is less than 0.5.

D. The normal distribution can be used because the original population has a normal distribution.

3. Probability
An accident at Eagle Aspirin Company led to bottles of their aspirin pain reliever being incorrectly manufactured. The 50 tablet bottles of aspirin were each given 45 pain reliever tablets and 5 placebo tablets by mistake. Suppose you have purchased a faulty bottle of Eagle aspirin pain reliever containing 50 tablets. You randomly select three tablets from the bottle to relieve your headache from studying statistics.
a.
What is the probability that all three tablets you selected are pain reliever tablets?
b.
What is the probability that all three tablets you selected are placebo tablets?
c.
What is the probability that exactly one of the three tablets you selected is a pain reliever tablet?
d.
What is the probability that at least one of the three tablets you selected is a pain reliever tablet?