In March 2025
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A manufacturer of isopropyl alcohol claims that their product has a mean content of 480 mL. and a standard
deviation of 21.5 mL. Assume that the variable is normally distributed.
a. If a sample is selected, what is the probability that the content will be less than 505 mL?
b. If a sample of 6 isopropyl alcohol is selected, what is the probability that the mean of the sample will be
less than 505?
Directions: Read the problems below and do what is asked. Write your answers on a separate sheet of paper.
1. The records of SCA Registrar show that the average final grade in Mathematics for STEM students is 91 with a standard deviation of 3. A group of student-researchers found out that the average final grade of 37 randomly selected STEM students in Mathematics is no longer 91. Use 0.05 level of significance to test the hypothesis and a sample mean within the range of 88 to 94 only.
A. State the hypotheses.
B. Determine the test statistic to use.
C. Determine the level of significance, critical value, and the decision rule.
D. Compute the value of the test statistic.
E. Make a decision.
F. Draw a conclusion.
2. An electrical company claims that the average life of the bulbs it manufactures is 1 200 hours with a standard deviation of 250 hours. If a random sample of 100 bulbs is chosen, what is the probability that the sample mean will be less than 1150 hours? (5 points)
a. Z = - 2
b. Z = - 3
c. Z = - 4
d. Z = - 5
Professors regularly give two versions of an exam. The professor may also provide summary statistics for each version. Suppose the following summary is provided:
A student who took Version A says that he should get an extra point because his exam was harder as evidenced by the lower mean score for Version A, as shown by the mean score. Does the student have a good argument? Pick the best answer below.
1 point
We need to know the minimum and the maximum for each version to determine if this argument is valid.
No. The average scores are relatively close when considering the spread of the distributions. The difference might just be due to just chance.
Yes. Only 53 students took exam Version A while 65 students took exam Version B.
We need to know the shape of the distribution for each version to determine if this argument is valid.
No. The median of Version A is higher.
Yes. The difference in the exam scores means that there is a difference in difficulty between the versions.
Based on the relative frequency histogram below, which of the following statements is supported by the plot?
1 point
It is not possible to estimate the median without knowing the sample size.
The IQR of the distribution is roughly 10.
There are no outliers in the distribution.
The distribution is multimodal.
The mean of the distribution is smaller than its median.
4.
Question 4
The midrange is defined as the average of the maximum and the minimum.
True or False: This statistic is robust to outliers.
1 point
False
True
5.
Question 5
It is relatively common for fish to be mislabeled in supermarkets and even in restaurants. The table below shows the results of a study where a random sample of 156 fish for sale were collected and genetically tested. The researchers classified each sample as being labeled properly or being mislabeled. What fraction of smoked fish in the sample were mislabeled? Choose the closest answer.
1 point
9%
78%
18%
72%
28%
A population consists of values (1, 4, 7). Consider all possible samples of size n= 3 that can be drawn without replacement from this population.
a. Find the mean of the population.
b. Find the standard deviation of the population.
c. Find the mean of the sampling distribution of means.
d. Find the standard deviation of the sample distribution of means,
e.Construct the probability histogram of x with replacement
List the criteria that u can use to evaluate the normality of a distribution .
Suppose that the continuous random variable has a Gamma distribution with shape parameter α and scale parameter β, that is X ∼ Γ(α, β) .
(a) Show that the moment generating function of the random variable X is Mx(t) = (1 −t/β)^−α ,
where t < β and use the result to derive the expectation and variance of X.
(b) Hence show that if (x1, · · · , xn) are independent random variable such that each has a Gamma distribution with parameter P αi and β where i = 1, 2, 3, · · · , n , then the random variable Y =sum n, i=1 Xi , has a Gamma distribution with parameter Sum n, i=1 αi and β.
(c) Deduce from your results in (ii) that if (x1, x2, · · · , xn) are iid exponential distributed with
parameter β, then Sn = Sum Xi ∼ Γ(n, β).
a) Let X, Y and Z be three independent exponential random variables with respectively means 1/λ1 , 1/λ2 and 1/λ3
. Find the P(X < Y < Z).
(b) Given that a random variable X has a Poisson distribution with parameter λ. Find E[cos(πx)],
if E(x) = ln 2.
a) A bivariate continuous random variable (U, V ) has join density function g(u, v), where −∞ <
u, v < ∞. If X = U − V and Y = V + U, show that the joint density function of X and Y is given by
f(x, y) = 1/2 g (x − y/2 , x + y/2) ,
and state a form of this result when U and V are independent.
(b) Suppose U and V are independent random variables with common uniform distribution over the interval [θ −1/4, θ +1/4]. Let X = U −V and Y = U +V . Find the joint density function of (X, Y ).
Hence find the marginal of X and Y .
