Jeremiah Mbaria

University of Nairobi
Math and Physics Teacher

Biography

I am a highly motivated team player with over eight years of experience having published work in
mathematics designed to engage and equip the students intellectually. My goal is to shape a better world for
the students by empowering them to realize their full potential.

Education

BA Statistics
University of Nairobi
MS Statistics
University of Nairobi

Educator Statistics

Numerade tutor for 6 years
2027 Students Helped

Topics Covered

Understanding Discrete Random Variables: A Comprehensive Guide
Maximizing Accuracy with Effective Sampling and Data Analysis
Unlocking Insights with Descriptive Statistics: A Comprehensive Guide
Linear Regression & Correlation: Analyzing Data Relationships
Hypothesis Testing with One Sample: A Comprehensive Guide
Mastering Analysis of Variance: Techniques for Accurate Results
Unlocking Insights with Non-Parametric Statistics | Boost Your Analysis
Testing Differences: Means, Proportions & Variances
Exploring Probability Topics: From Basics to Advanced Strategies
Unlocking the Power of Confidence Intervals: A Comprehensive Guide
Mastering Equations and Inequalities: Your Guide to Mathematical Success
Solving Systems of Equations and Inequalities: A Comprehensive Guide
Hypothesis Testing with Two Samples: A Comprehensive Guide
Master Probability and Counting Rules for Better Outcomes
Understanding the Normal Distribution: A Comprehensive Guide
Understanding Probability and Statistics: Key Concepts and Principles
Understanding Confidence Intervals and Sample Size

Jeremiah's Textbook Answer Videos

13:11
Probability with Applications in Engineering, Science, and Technology

A study of children's intelligence and behavior included the following IQ data for 33 first-graders
that participated in the study.
$\begin{array}{ccccc}{82} & {96} & {99} & {102} & {103} & {103} \\ {108} & {109} & {110} & {110} & {111} & {113} \\ {118} & {118} & {119} & {121} & {122} & {122}\end{array}$
$\begin{array}{lllll}{106} & {107} & {108} & {108} & {108} \\ {113} & {113} & {113} & {115} & {115} \\ {127} & {132} & {136} & {140} & {146}\end{array}$
(a) Calculate a point estimate of the mean IQ for the conceptual population of all first graders in
this school, and state which estimator you used.
(b) Calculate a point estimate of the IQ value that separates the lowest 50$\%$ of all such students
from the highest $50 \%,$ and state which estimator you used.
(c) Calculate and interpret a point estimate of the population standard deviation $\sigma .$ Which
estimator did you use?
(d) Calculate a point estimate of the proportion of all such students whose IQ exceeds 100.
(e) Calculate a point estimate of the population coefficient of variation, $100 \sigma / \mu,$ and state what estimator you used.

Chapter 5: The Basics of Statistical Inference
Section 1: Point Estimation
Jeremiah Mbaria
21:57
Probability with Applications in Engineering, Science, and Technology

The data set mentioned in Exercise 1 also includes these third-grade IQ observations for males:
$\begin{array}{lll}{117} & {103} & {121} \\ {149} & {125} & {131}\end{array}$
$\begin{array}{lll}{112} & {120} & {132} \\ {136} & {107} & {108}\end{array}$
$\begin{array}{lll}{113} & {117} & {132} \\ {113} & {136} & {114}\end{array}$
and females:
$\begin{array}{lll}{114} & {102} & {113} \\ {114} & {109} & {102}\end{array}$
$\begin{array}{ll}{131} & {124} \\ {114} & {127}\end{array}$
$\begin{array}{ll}{117} & {120} \\ {127} & {103}\end{array}$ 90
Prior to obtaining data, denote the male values by $X_{1}, \ldots, X_{m}$ and the female values by $Y_{1}, \ldots, Y_{n}$
Suppose that the $X_{i}$ s constitute a random sample from a distribution with mean $\mu_{1}$ and standard
deviation $\sigma_{1}$ and that the $Y_{i}$ s form a random sample (independent of the $X_{i}$ s ) from another
distribution with mean $u_{2}$ and standard deviation $\sigma_{2}$ .
(a) Show that $\overline{X}-\overline{Y}$ is an unbiased estimator of $\mu_{1}-\mu_{2}$ . Then calculate the estimate for the
given data.
(b) Use rules of variance from Chap. 4 to obtain an expression for the standard error of the
estimator in (a), and then compute the estimated standard error.
(c) Calculate a point estimate of the ratio $\sigma_{1} / \sigma_{2}$ of the two standard deviations.
(d) Suppose one male third-grader and one female third-grader are randomly selected. Calculate
a point estimate of the variance of the difference $X-Y$ between their IQs.

Chapter 5: The Basics of Statistical Inference
Section 1: Point Estimation
Jeremiah Mbaria
06:28
Probability with Applications in Engineering, Science, and Technology

Consider the accompanying observations on stream flow (thousands of acre-feet) recorded at a
station in Colorado for the period April $1-$ August 31 over a $31-$ year span (from an article in the
1974 volume of Water Resources Res.).
127.96
285.37
200.19
125.86
117.64
204.91
94.33
210.07
100.85
66.24
114.79
302.74
311.13
203.24
89.59
247.11
109.11
280.55
150.58
108.91
185.36
299.87
330.33
145.11
262.09
178.21
126.94
109.64
85.54
95.36
477.08
An appropriate probability plot supports the use of the lognormal distribution ( see Sect. 3.5$)$ as a
reasonable model for stream flow.
(a) Estimate the parameters of the distribution. [Hint: Remember that $X$ has a lognormal
distribution with parameters $\mu$ and $\sigma$ if $\ln (X)$ is normally distributed with mean $\mu$ and
standard deviation $\sigma . ]$
(b) Use the estimates of part (a) to calculate an estimate of the expected value of stream flow
$\quad$ [Hint: What is the expression for $E(X) ? ]$

Chapter 5: The Basics of Statistical Inference
Section 1: Point Estimation
Jeremiah Mbaria
27:31
Probability with Applications in Engineering, Science, and Technology

Let $X_{1}, \ldots, X_{n}$ be a random sample from a distribution with mean $\mu$ and variance $\sigma^{2}$ .
(a) Show that $\sum\left(X_{i}-\overline{X}\right)^{2}=\left(\sum X_{i}^{2}\right)-n \overline{X}^{2}$
(b) Show that $E\left(\sum X_{i}^{2}\right)=n\left(\mu^{2}+\sigma^{2}\right) .[$Hint$:$ Use linearity of expectation, along with the relation
$E\left(Y^{2}\right)=\operatorname{Var}(Y)+[E(Y)]^{2} \cdot ]$
(c) Show that $E\left(n \overline{X}^{2}\right)=n \mu^{2}+\sigma^{2} .[H i n t :$ Apply the relation given in the previous hint, but this
$\quad$ time to $Y=\overline{X} . ]$
(d) Combine parts $(\mathrm{a})-(\mathrm{c})$ to show that $S^{2}$ is an unbiased estimator of $\sigma^{2} .$
(e) Does it follow that the sample standard deviation, $S,$ of a random sample is an unbiased
estimator of $\sigma ?$ Why or why not?

Chapter 5: The Basics of Statistical Inference
Section 1: Point Estimation
Jeremiah Mbaria
07:54
Probability with Applications in Engineering, Science, and Technology

Let $X_{1}, X_{2}, \ldots, X_{n}$ represent a random sample from a Rayleigh distribution with pdf
$f(x ; \theta)=\frac{x}{\theta} e^{-x^{2} /(2 \theta)} \quad x>0$
(a) It can be shown that $E\left(X^{2}\right)=2 \theta .$ Use this fact to construct an unbiased estimator of $\theta$ based
$\quad$ on $\sum X_{i}^{2}$ (and use rules of expected value to show that it is unbiased).
(b) Estimate $\theta$ from the following measurements of blood plasma beta concentration
$\quad($ in $\operatorname{pmol} / L)$ for $n=10$ men.
$\begin{array}{ll}{16.88} & {10.23} \\ {14.23} & {19.87}\end{array}$
$\begin{array}{ll}{4.59} & {6.66} \\ {9.40} & {6.51}\end{array}$
13.68
10.95

Chapter 5: The Basics of Statistical Inference
Section 1: Point Estimation
Jeremiah Mbaria
15:23
Probability with Applications in Engineering, Science, and Technology

A reservation service employs five information operators who receive requests for information independently of one another, each according to a Poisson process with rate $\lambda=2$ per minute.
(a) What is the probability that during a given 1 -min period, the first operator receives no requests?
(b) What is the probability that during a given 1 -min period, exactly four of the five operators receive no requests?
(c) Write an expression for the probability that during a given 1 -min period, all of the operators receive exactly the same number of requests.

Chapter 2: Discrete Random Variables and Probability Distributions
Section 9: Supplementary Exercises
Jeremiah Mbaria
1 2 3 4 5 ... 75

Jeremiah's Quick Ask Videos

01:36
Precalculus

½ log n + 3 log m
single and simplify if possible

Jeremiah Mbaria
03:29
Physics 101 Mechanics

A garden house has a cross sectional area of 1.35x10-4 m2. Inside the hose, the water flow with a speed of 1.46 m/s. If the cross section area of the nozzle was 0.43x10-4 m2, then determine the speed of water in the nozzle.

Jeremiah Mbaria
06:00
Physics 101 Mechanics

Only 2 forces act on a 7.4 kg body that can move over a
frictionless horizontal floor. One force is 9.7 N acting due east,
and the other force is 12.4 N acting 30 degrees north of east. What
is the magnitude of the body's acceleration? Hint: Acceleration
will only occur in the x-direction.

Jeremiah Mbaria
03:57
Geometry

The head of a steel meat tenderizer is a cube with a side length
of 5 centimeters. One face of the cube contains 25 square pyramids
that protrude from the surface. The side length of the base of each
square pyramid is .5 centimeter, and the height of each .5
centimeter. The mass of the head of the tenderizer is 1000 grams.
What is the density of the steel ? Roud to the nearest hundredth.
Show your work.

Jeremiah Mbaria
05:00
Geometry

A solid spherical steel ball 20 cm in diameter is placed into a
tall vertical cylinder containing water, causing the water level to
rise by 10 cm. What is the radius of the cylinder?

Jeremiah Mbaria
09:41
Geometry

An airplane has an airspeed of 580 km/h bearing 49 degrees north of east. The wind velocity is 40 km/h in the direction 32 degrees north of west. Find the resultant velocity representing the path of the airplane with respect to the ground.
What is the actual ground speed of the aircraft? (Do not round until the final answer. Then round to the nearest tenth as needed.)

Jeremiah Mbaria
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