Data Distribution Questions and Answers

We are interested in significant differences (level ${\displaystyle \alpha =0.05}$) between the expected values ${\displaystyle {\mu }_1\text{and}{\mu }_2}$ of two populations. Which of the following statements on statistical tests are correct?
a) We will formulate the null hypothesis as ${\displaystyle {\mu }_1={\mu }_2}$.
b) A t-test can always be applied in this situation.
c) A p-value is the probability that the null hypothesis is correct, given the observed data.
d) If we obtain a p-value of 0.04, we will reject (level ${\displaystyle \alpha =0.05}$) the null hypothesis.

An environmentalist wants to find out the fraction of oil tankers that have spills each month.
Suppose a sample of 273 tankers is drawn. Of these ships, 194 did not have spills. Using the data, construct the 85% confidence interval for the population proportion of oil tankers that have spills each month. Round your answers to three decimal places.

Following data reveals the usage of units of electricity by different consumers of a certain area:
$$\begin{array}{|ccccccc|}\hline \text{Units consumed}& 1-50& 51-100& 101-150& 151-200& 201-250& 251-300\\ \text{Number of Users(Frequency)}& 24& 68& 104& 126& 84& 44\\ \hline\end{array}$$
Find the following for given data
1. Arithmetic Mean.
2. Median
3. All Quartiles

An engineering company has four openings and the applicant pool consists of 6 database administrators and eight network engineers. All are equally qualified so the hiring will be done randomly. The hiring committee consists of 4 women and 4 men.
a. If one person on the hiring committee is chosen at random to draw the names out of a hat, what is the probability that the person drawing the names is a woman.
b. How many ways can the group who are to be hired be formed if they are no restrictions on composition?
c. How many ways can 3 database administrators be chosen?
d. How many ways can 1 network engineers be chosen?
e. What is the probability that the random selection of the four persons to be hired will result in 3 database administrators and 1 network engineer?

The following table gives the scores of 30 students in a mathematics examination.
$$\begin{array}{|cccccc|}\hline Scores& 90-99& 80-89& 70-79& 60-69& 50-59\\ Students& 3& 8& 15& 3& 1\\ \hline\end{array}$$
Find the mean and the standard deviation of the given data. Hint: Assume that all scores lying within a group interval take the middle value of that group. (Round your answers to two decimal places.)
mean
standard deviation

Of the trains that recently arrived at Hampton Station, 4 were on time and 2 were late. Considering this data, how many of the next 12 trains would you expect to be on time?
? trains

The following data represents the number of home runs that Ryan Howard, a major league baseball player for the Philadelphia, hit over a six-year period.
22 58 47 48 45 31
what is the median for this data set?

Suppose, household color TVs are replaced at an average age of ${\displaystyle \mu =8.2years}$ after purchase, and the (95% of data) range was from 6.2 to 10.2 years. Thus, the range was ${\displaystyle 10.2-6.2=4.0}$ 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 symmetrical 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.)

Calculate the Mean, Median and Modal numbers of persons per house from the data:
$$\begin{array}{|ccccccccccc|}\hline \text{No. of persons per house}& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10\\ \text{No. of houses}& 26& 113& 120& 95& 60& 42& 21& 14& 5& 4\\ \hline\end{array}$$

Based on historical data, on average, Florida experiences one flooding event every 2.4 months. Assume that the time between flooding events can be modeled using an exponential distribution. What is the median number of months from now until Florida experiences its next flooding event?

A recent claim has been made that people who have iPads spend more time on their iPads than people who have laptops spend on their laptops. After all calculations were performed, the study notes the z statistic to be -2.81 compared to significance level of ${\displaystyle \alpha =0.01}$.
Is there truly a significant difference between the two data sets?
Since, p-value ? the significance level, ${\displaystyle \alpha }$, we can conclude that there ? a truly significant difference between the two data sets.

$$\begin{array}{|ccc|}\hline \text{Lower Class Limit}& \text{Upper Class Limit}& \text{Frequency}\\ 30& 34& 1\\ 35& 39& 1\\ 40& 44& 2\\ 45& 49& 6\\ 50& 54& 5\\ 55& 59& 11\\ 60& 64& 10\\ 65& 69& 21\\ 70& 74& 13\\ \hline\end{array}$$
What is the mean for the following Grouped Frequency Data Table (GFDT).
To save time, copy the data in MOM and paste special (as text) into the ESP3 Excel sheet. You can then copy the answer from Excel and paste it into the answer box on MOM.

A local manufacturer of copying equipment includes a repair warranty with each copier. Virtually all of his equipment exhibit an increasing failure rate. Based on historical repair data, the failure rate for Model 25cc7 is accurately described by the function ${\displaystyle r\left(t\right)=2.7786t^{1.3}}$, where t is measured in months of continuous operation. What is the probability that the time between two successive failures of this equipment exceeds two months of operation?

A university asked its applicants to write a short essay for its entrance test. Based on their gathered data, it was found out that two out 10 students did not make grammatical mistakes, three out of 10 students made a grammatical mistake, four out of 10 students made two mistakes, and one out of 10 made three grammatical mistakes. What is the average number of grammatical mistakes made by the students in the essay?

The testing division of a chip manufacturing company tests all manufactured chips individually, one at a time. Based on the company data, with probability 0.9, the first defective chip tested on a given day appears after the 10th tested chip. Find the probability that the first defective chip appears between the 20th and the 30th tested chips excluding the 20th and the 30th chips on a given day.

An insurance policy sells for $600. Based on past data, an average of 1 in 50 policyholders will file a $5,000 claim, an average of 1 in 100 policyholders will file a $10,000 claim, and 1 in 200 policyholders will file a $30,000 claim. Find the expected value (to the company) per policy sold. If the company sells 10,000 policies, what is the expected profit or loss? Explain

Based on historical data, your manager believes that 35% of the company's orders come from first-time customers. A random sample of 175 orders will be used to estimate the proportion of first-time-customers. What is the probability that the sample proportion is between 0.21 and 0.39?