Master Statistics and Probability Problems with Expert Help

In order to use a normal distribution to compute confidence intervals for p, what conditions on np and nq need to be satisfied?

Assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance ${\displaystyle {\sigma }^2}$ and (b) the population standard deviation σ. Interpret the results. The reserve capacities (in hours) of 18 randomly selected automotive batteries have a sample standard deviation of 0.25 hour. Use an 80% level of confidence.

1. The standard error of the estimate is the same at all points along the regression line because we assumed that A. The observed values of y are normally distributed around each estimated value of y-hat. B. The variance of the distributions around each possible value of y-hat is the same. C. All available data were taken into account when the regression line was calculated. D. The regression line minimized the sum of the squared errors. E. None of the above.

For a test of $H_0:p=0.5,$ the z test statistic equals 1.74. Find the p-value for $H_a:p>0.5$.
a) 0.0446
b) 0.0409
c) 0.892
d) 0.9591
e) 0.0818
f) 0.9554

1. Who seems to have more variability in their shoe sizes, men or women?
a) Men
b) Women
c) Neither group show variability
d) Flag this Question
2. In general, why use the estimate of $n-1$ rather than n in the computation of the standard deviation and variance?
a) The estimate n-1 is better because it is used for calculating the population variance and standard deviation
b) The estimate n-1 is never used to calculate the sample variance and standard deviation
c) $n-1$ provides an unbiased estimate of the population and allows more variability when using a sample and gives a better mathematical estimate of the population
d) The estimate n-1 is better because it is use for calculation of both the population and sample variance as well as standard deviation.
$\begin{array}{|cc|}\hline \text{Shoe Size (in cm)}& \text{Gender (M of F)}\\ 25.7& M\\ 25.4& F\\ 23.8& F\\ 25.4& F\\ 26.7& M\\ 23.8& F\\ 25.4& F\\ 25.4& F\\ 25.7& M\\ 25.7& F\\ 23.5& F\\ 23.1& F\\ 26& M\\ 23.5& F\\ 26.7& F\\ 26& M\\ 23.1& F\\ 25.1& F\\ 27& M\\ 25.4& F\\ 23.5& F\\ 23.8& F\\ 27& M\\ 25.7& F\\ \hline\end{array}$
$\begin{array}{|cc|}\hline \text{Shoe Size (in cm)}& \text{Gender (M of F)}\\ 27.6& M\\ 26.9& F\\ 26& F\\ 28.4& M\\ 23.5& F\\ 27& F\\ 25.1& F\\ 28.4& M\\ 23.1& F\\ 23.8& F\\ 26& F\\ 25.4& M\\ 23.8& F\\ 24.8& M\\ 25.1& F\\ 24.8& F\\ 26& M\\ 25.4& F\\ 26& M\\ 27& M\\ 25.7& F\\ 27& M\\ 23.5& F\\ 29& F\\ \hline\end{array}$

Describe how to use random numbers to simulate the following PERFORMANCES:
(a) A basketball player has the ABILITY to make 40% of his shots and he takes 25 shots in a game.
(b) A basketball player has the ABILITY to make 75.2% of her free-throws and she takes 8 free throws in a game.

For each of the following confidence intervals for the difference between two means,
$({\mu }_1-{\mu }_2)$
which mean is significantly larger?
a.(-10.5)

b.(-10,-5)

c.(5,10)

Chances: Risk and Odds in Everyday Life, by James Burke, reports that only 2% of all local franchises are business failures. A Colorado Springs shopping complex has 137 franchises (restaurants, print shops, convenience stores, hair salons, etc.). Let r be the number of these franchises that are business failures. Explain why a Poisson approximation to the binomial would be appropriate for the random variable r. What is n? What is p? What is λλ (rounded to the nearest tenth)?

Complete the two-way table.

 $\begin{array}{|cccc|}\hline \text{ Gender }& \text{ Response }& & \\ & \text{ Yes }& \text{ No }& \text{ Total }\\ \text{ Male }& & 52& \\ \text{ Female }& 48& & 110\\ \text{ Total }& 65& & \\ \hline\end{array}$

$\overline{\begin{array}{cc}& \text{Afraid to walk at night?}\\ & \text{Yes}& \text{No}& \text{Total}\\ \text{Male}& 173& 598& 771\\ \text{Female}& 393& 540& 933\\ \text{Total}& 566& 1138& 1704\\ \text{Source:}2014GSS\end{array}}$
If the chi-square $(\chi 2)$ test statistic $=73.7$ what is the p-value you would report? Use Table C.Remember to calculate the df first.
Group of answer choices
$P>0.250$
$P=0.01$
$P<0.001$
$P<0.002$

$\begin{array}{|cc|}\hline & \text{Right-Tail Probability}\\ df& 0.250& 0.100& 0.050& 0.025& 0.010& 0.005& 0.001\\ 1& 1.32& 2.71& 3.84& 5.02& 6.63& 7.88& 10.83\\ 2& 2.77& 4.61& 5.99& 7.38& 9.21& 10.60& 13.82\\ 3& 4.11& 6.25& 7.81& 9.35& 11.34& 12.84& 16.27\\ 4& 5.39& 7.78& 9.49& 11.14& 13.28& 14.86& 18.47\\ 5& 6.63& 9.24& 11.07& 12.83& 15.09& 16.75& 20.52\\ 6& 7.84& 10.64& 12.59& 14.45& 16.81& 18.55& 22.46\\ 7& 9.04& 12.02& 14.07& 16.01& 18.48& 20.28& 24.32\\ 8& 10.22& 13.36& 15.51& 17.53& 20.09& 21.96& 26.12\\ 9& 11.39& 14.68& 16.92& 19.02& 21.67& 23.59& 27.88\\ 10& 12.55& 15.99& 18.31& 20.48& 23.21& 25.19& 29.59\\ 11& 13.70& 17.28& 19.68& 21.92& 24.72& 26.76& 31.26\\ 12& 14.85& 18.55& 21.03& 23.34& 26.22& 28.30& 32.91\\ 13& 15.98& 19.81& 22.36& 24.74& 27.69& 29.82& 34.53\\ 14& 17.12& 21.06& 23.68& 26.12& 29.14& 31.32& 36.12\\ 15& 18.25& 22.31& 25.00& 27.49& 30.58& 32.80& 37.70\\ 16& 19.37& 32.54& 26.30& 28.85& 32.00& 34.27& 39.25\\ 17& 20.49& 24.77& 27.59& 30.19& 33.41& 35.72& 40.79\\ 18& 21.60& 25.99& 28.87& 31.53& 34.81& 37.16& 42.31\\ 19& 22.72& 27.20& 30.14& 32.85& 36.19& 38.58& 43.82\\ 20& 23.83& 28.41& 31.41& 34.17& 37.57& 40.00& 45.32\\ \hline\end{array}$

Can filling car tires with nitrogen instead of compressed air increase gas mileage? Describe how you would design a study to try to answer this question. Be as specific as you can about the details of your study.

Here’s an interesting challenge you can give to a friend. Hold a $1 (or larger!) bill by an upper corner. Have a friend prepare to pinch a lower corner, putting her fingers near but not touching the bill. Tell her to try to catch the bill when you drop it by simply closing her fingers. This seems like it should be easy, but it’s not. After she sees that you have released the bill, it will take her about 0.25 s to react and close her fingers-which is not fast enough to catch the bill. How much time does it take for the bill to fall beyond her grasp? The length of a bill is 16 cm.

Water flows through a water hose at a rate of $Q_1=680c{m}^3/s$, the diameter of the hose is $d_1=2.2cm$. A nozzle is attached to the water hose. The water leaves the nozzle at a velocity of $v_2=9.2m/s$.
a) Enter an expression for the cross-sectional area of the hose, $A_1$, in terms of its diameter, $d_1$
b) Calculate the numerical value of $A_1,$ in square centimeters.
c) Enter an expression for the speed of the water in the hose, $v_1$, in terms of the volume floe rate $Q_1$ and cross-sectional area $A_1$
d) Calculate the speed of the water in the hose, $v_1$ in meters per second.
e) Enter an expression for the cross-sectional area of the nozzle, $A_2$, in terms of $v_1,v_2$ and $A_1$
f) Calculate the cross-sectional area of the nozzle, $A_2$ in square centimeters.

The following table represents the Frequency Distribution and Cumulative Distributions for this data set: 12, 13, 17, 18, 18, 24, 26, 27, 27, 30, 30, 35, 37, 41, 42, 43, 44, 46, 53, 58

$\begin{array}{|cccc|}\hline \text{Class}& \text{Frequency}& \text{Relative Frequency}& \text{Cumulative Frequency}\\ \text{10 but les than 20}& 5\\ \text{20 but les than 30}& 4\\ \text{30 but les than 40}& 4\\ \text{40 but les than 50}& 5\\ \text{50 but les than 60}& 2\\ \text{TOTAL}\\ \hline\end{array}$

What is the Relative Frequency for the class: 20 but less than 30? State you answer as a value with exactly two digits after the decimal. for example 0.30 or 0.35

When a correlation value is reported in research journals, there often is not an accompanying scatterplot. Explain why reported correlation values should be supported with either a scatterplot or a description of the scatterplot.

Suppose you were to collect data for the pair of variables. You want to make a scatterplot. Which variable would you use as the explanatory variable and which as the response variable? Why? What would you expect to see in the scatterplot? Discuss the likely direction, form, and strength. College freshmen: shoe size, grade point average

The file sapphire lists observed values of Young’s modulus (g) measured at various temperatures (T ) for sapphire rods (Ku 1969). Fit a linear relationship ${\displaystyle g={\beta }_0+{\beta }_{1}t}$, and form confidence intervals for the coefficients. Examine the residuals.

Smoking and drinking coffee have a tendency to stain teeth. In an effort to determine the ability of chewing gum to remove stains on teeth, researchers conducted an experiment in which 64 bovine incisors (teeth) were stained with natural pigments such as coffee for 10 days. Each tooth was randomly assigned to one of four treatments: gum A, gum B, gum C, or saliva. Ea ch tooth group was placed into a device that simulated a human chewing gum. The temperature of the device was maintained at body temperature and the tooth was in the device for 20 minutes. The process was repeated six times (for a total of 120 minutes of chewing). The researcher conducting the experiment did not know which treatment was being applied to each experimental unit. Upon removing a tooth from the chewing apparatus, the color change was measured using a spectrophotometer. The percent age of stain removed by each treatment after 120 minutes is as follows: Gum A, 47.6%; Gum B, 45.2%, Gum C, 21.4%, Saliva, 2.1%. The researchers concluded that gums A and B removed significantly more stain than gum C or saliva. In addition, gum C removed significantly more stain than saliva. (a) Identify the research objective. (b) Is this an observational study or designed experiment? Why? (c) If observational, what type of observational study is this? If an experiment, what type of experimental design is this? (d) What is the response variable? (e) What is the explanatory variable? ls it qualitative or quantitative? (I) Identify the experimental units. (g) State a factor that could affect the value of the response variable that is fixed at a set level. (h) What is the conclusion of the study?

The two-way table below describes the members of the U.S Senate in a recent year.
Male Female
Democrats 47 13
Republicans 36 4
If we select a U.S. senator at random, what's the probability that the senator is a female and a Democrat?

Let X and Y be independent, continuous random variables with the same maginal probability density function, defined as
$f_X(t)=f_Y(t)=\{\begin{array}{l}\frac{2}{t^2},t>2\\ 0,otherwise\end{array}$
(a)What is the joint probability density function f(x,y)?
(b)Find the probability density of W=XY. Hind: Determine the cdf of Z.