Data Distribution Questions and Answers

Suppose, household color TVs are replaced at an average age of ${\displaystyle \mu =8.4years}$ after purchase, and the (95% of data) range was from 4.4 to 12.4 years. Thus, the range was ${\displaystyle 12.4-4.4=8.0}$ years. Let x be the age (in years) at which a color TV is replaced. Assume that x has a distribution that is approximately normal.
(a) The empirical rule indicates that for a symmetric and bell-shaped distribution, approximately 95% of the data lies within two standard deviations of the mean. Therefore, a 95% range of data values extending from ${\displaystyle \mu -2\sigma \to \mu +2\sigma }$ is often used for "commonly occurring" data values. Note that the interval from ${\displaystyle \mu -2\sigma \to \mu +2\sigma is4\sigma }$ in length. This leads to a "rule of thumb" for estimating the standard deviation from a 95% range of data values.
Estimating the standard deviation
For a symmetric, bell-shaped distribution,
${\displaystyle \text{standard deviation}\approx ran\ge 4\approx \text{high value}-\text{low value}}$
where it is estimated that about 95% of the commonly occurring data values fall into this range.
Use this "rule of thumb" to approximate the standard deviation of x values, where x is the age (in years) at which a color TV is replaced. (Round your answer to one decimal place.)
(b) What is the probability that someone will keep a color TV more than 5 years before replacement? (Round your answer to four decimal places.)
(c) What is the probability that someone will keep a color TV fewer than 10 years before replacement? (Round your answer to four decimal places.)

March Madness Movies served 23 lemonades out of a total of 111 fountain dri
last weekend.
Based on this data, what is a reasonable estimate of the probability that the next fountain drink ordered is a lemonade?

Find the range and standard deviation of the set of data
210,215,220,225,230,235,240
The range is 30.
The standard deviation is ?

The honey farm has installed a filling machine for honey jars that hold 500 g of honey. Doris is calibrating the new machine. She sets the machine to a mean of 500 g and performs a test run of 48 jars. The table attached to the quiz post in the Google Classroom shows the results. Assume it is normally distributed.
Determine the mean of the data
Determine the standard deviation of the data.
What is the probability that a jar contains at least 504 g of honey?

Analyze the following data. Assume the true value of L (the length of a table) is 93.40cm.
$$\begin{array}{|cccccccc|}\hline L(cm)& 93.2& 93.3& 93.9& 92.8& 93.5& 93.4& 92.9\\ \hline\end{array}$$
a) What are the mean and standard deviation of this data sample?
b) What is the uncertainty of the mean?
c) Report the measured value of L.

The following table gives the scores of 30 students in a mathematics examination:
$$\begin{array}{|cccccc|}\hline Scores& 90-99& 80-89& 70-79& 60-69& 50-59\\ Students& 4& 8& 12& 4& 2\\ \hline\end{array}$$
Find the mean and the standard deviation of the distribution of the given data.

In a survey of daily internet game time usage by college school students, a sample of 7 students provided the following data:
3,1.5,2.2,1.5,0.5,0,2.5
Find the mean of the data.

If the mean is 25 for the data:
20,30,15,24,y,8
find the median
1) 22
2) 53
3) 20
4) 24

Some non-linear regressions can also be estimated using a linear regression model (using "linearization"). Assume that the data below show the selling prices y (in dollars) of a certain equipment against its age x (in years). We'd like to fit a non-linear regression im the form ${\displaystyle \hat{y}=c{d}^X}$ to estimate parameters c and d from the data by linearizing the model through ${\displaystyle \ln \hat{y}=\ln c+\left(\ln d\right)x=b_0+b_{1}x}$.
$$\begin{array}{|cccc|}\hline x& y& x& y\\ 1& 6312& 3& 5387\\ 2& 5697& 5& 4973\\ 2& 5734& 5& 4892\\ \hline\end{array}$$
Using Excel ot other software, the non-linear regression model ${\displaystyle \hat{y}=c{d}^X}$ can be estimated as ${\displaystyle \hat{y}=?{(?)}^X}$

Consider the boxplot below.
43 58 61 67 74
a. What quarter has the smallest spread of data?
- Third
- First
- Fourth
- Second
b. What is that spread?
c. What quarter has the largest spread of data?
- Fourth
- Second
- First
- Third

In an orchard, harvested apples are randomly distributed in packets, each containing 7 fruits.
The orchard had a total of 25 apples collected, and 10 fruits were considered of excellent quality. Let X be the number of optimal apples allocated to first package, which of the probability distributions would be adequate to describe the random variable X?
(a) Geometric ${\displaystyle \left(\frac{7}{25}\right)}$
(b) Binomial ${\displaystyle (7,\frac{3}{25})}$
(c) Poisson (3)
(d) Geometric ${\displaystyle \left(\frac{3}{25}\right)}$
(e) Hypergeometric (25,10,7)

Find the equation of the regression line for the given data. Round values to the thousandth
$$\begin{array}{|ccccccccccc|}\hline x& -5& -3& 4& 1& -1& -2& 0& 2& 3& -4\\ y& 11& 6& -6& -1& 3& 4& 1& -4& -5& 8\\ \hline\end{array}$$
1) ${\displaystyle \bar{y}=1.885x-0.755}$
2) ${\displaystyle \bar{y}=-0.758x-1.885}$
3) ${\displaystyle \bar{y}=-1.885x+0.758}$
4) ${\displaystyle \bar{y}=0.758x+1.885}$

For the following Grouped Frequency Data Table (GFDT) answer the questions below.
$$\begin{array}{|ccc|}\hline \text{Lower Class limit}& \text{Upper Class Limit}& \text{Frequency}\\ 30& 34& 10\\ 35& 39& 9\\ 40& 44& 4\\ 45& 49& 10\\ 50& 54& 7\\ 55& 59& 8\\ 60& 64& 8\\ 65& 69& 6\\ 70& 74& 14\\ \hline\end{array}$$
(a) What is the 88-th percentile for the following Grouped Frequency Data Table?
(b) What is the 9-th decile for the following Grouped Frequency Data Table?
(c) What is the median for the following Grouped Frequency Data Table?

A lamp holds 3 bulbs. There are two kinds of bulbs, short-lived red ones with mean lifetime 1 day, and long-lived green ones with mean lifetime 3 days. All bulb lifetimes have exponential distributions independent of each other. Whenever a green light bulb burns out, it is replaced by a red one. Whenever a red light bulb burns out, it is replaced by a green one. Define variables and write equations that, if solved, would yield the probability distribution of the number of green light bulbs at a random moment when a bulb has just been replaced.

Explain Probability density function?

The observations given below are related with duration between two patients that come to ER service of certain hospital. There is a claim that the distribution of durations is Exponential with ${\displaystyle V\left(X\right)=1.778}$. Evaluate the claim.
$$\begin{array}{|ccccccccccc|}\hline 3.264& 4.62& 0.439& 0.13& 1.975& 1.271& 0.82& 0.42& 1.145& 0.61& 2.408\\ 0.87& 1.942& 0.32& 0.62& 0.375& 0.383& 2.512& 1.92& 3.19& 2.12& 3.89\\ \hline\end{array}$$

You are given:
(i) The annual number of claims on a given policy has the geometric distribution.
(ii) One-third of the policies have two claims ${\displaystyle (\beta =2)}$, abd the remaining two-thirds have five claims ${\displaystyle (\beta =5)}$,

A sensor connected to a control center indicates that there is damage somewhere along a particular 4-mile length of train track.
The conductor thi
that the exact location of the damage is uniformly distributed between mile 0 and mile 4 of that length of track. Let X be the distance along the track from mile 0 to the location of the damage.
a) Let f(x) be the probability density function of X. What is the value of f(1)?
b) What is the expected value of X?
c) What is the probability that the damage is located between mile 0 and mile 10 of that length of track?

Persons entering a blood bank are such that 1 in 9 have type O+ blood and 1 in 30 have type O- blood. Consider six randomly selected donors for the blood bank. Let X denote the number of donors with type O+ blood and Y denote the number with type O- blood. Find the probability distributions for X and Y. Also find the probability distribution for X+Y, the number of donors who have type O blood.