Mastering Statistics: Questions and Expert Solutions - Education

Welcome back, statistics enthusiasts! Today, we delve into the depths of statistical theory, exploring challenging questions that test your understanding and reasoning. Whether you're a student seeking to sharpen your skills or someone curious about the intricacies of statistics, this blog post is tailored just for you.

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Let's dive into the fascinating world of statistics with a couple of master-level theory questions, accompanied by comprehensive solutions provided by our expert.

Question 1: Probability Distributions

Consider a random variable $X$ with probability density function $f (x) = 3 1 x^{2}$ for $0 \leq x \leq 3$ , and zero otherwise.

a) Find the cumulative distribution function $F (x)$ for $X$ .

b) Determine the probability that $X$ takes a value less than or equal to $2$ .

Solution:

a) To find the cumulative distribution function $F (x)$ , we integrate the probability density function $f (x)$ from $- \infty$ to $x$ :

$F (x) = \int_{- \infty x} f (t) d t = \int_{0 x} 3 1 t^{2} d t$

$= 3 1 \cdot 3 t ^{3} ∣ ∣_{0 x} = 9 x ^{3}$

b) The probability that $X$ takes a value less than or equal to $2$ is given by $F (2)$ :

$F (2) = 9 2 ^{3} = 9 8$

Therefore, the probability that $X$ takes a value less than or equal to $2$ is $9 8$ .

Question 2: Hypothesis Testing

A company claims that the average weight of their cereal boxes is $500$ grams. A sample of $36$ cereal boxes is selected, and the sample mean weight is found to be $505$ grams, with a sample standard deviation of $10$ grams. Test the company's claim at a $5%$ level of significance.

Solution:

Given that the population standard deviation is unknown, we use a t-test for the mean.

The null hypothesis $H_{0}$ is that the average weight of the cereal boxes is $500$ grams.

The alternative hypothesis $H_{1}$ is that the average weight of the cereal boxes is not $500$ grams.

The test statistic for the t-test is given by:

$t = s / n x ˉ - μ$

where $x ˉ$ is the sample mean, $μ$ is the population mean under the null hypothesis, $s$ is the sample standard deviation, and $n$ is the sample size.

Plugging in the values:

$36 505 - 500 = 10/6 5 = 3$

With $35$ degrees of freedom (since $n - 1 = 35$ ), the critical value for a $5%$ level of significance is approximately $\pm 2.03$ (two-tailed test).

Since $∣ t ∣ = 3 > 2.03$ , we reject the null hypothesis.

Therefore, there is sufficient evidence to conclude that the average weight of the cereal boxes is not $500$ grams at the $5%$ level of significance.

In this blog post, we've explored two challenging statistics theory questions and provided detailed solutions. Remember, if you ever find yourself struggling with similar questions or need assistance with your statistics test, you can always rely on LiveExamHelper.com. Don't hesitate to reach out and say, "I want to pay someone to do my statistics test," and our expert team will be ready to assist you every step of the way.

Keep practicing, stay curious, and embrace the world of statistics with confidence!