# In congruence classes Z/(mZ), reduce the equation a_m*x_m^2=c_m either by finding convenient representation for a_m and b_m or by using the inverse of a_m. Then find a solution for this congruence directly or by replacing c_m : with its appropriate representative in Z/(mZ). If there is no solution explain why. Here a_m, b_m, x_m(=x),c_m in Z/(mZ): In Z/(19Z), [2]*x^2=[13]:

Question
Congruence
In congruence classes $$\displaystyle\frac{{Z}}{{{m}{Z}}}$$, reduce the equation $$\displaystyle{a}_{{m}}\cdot{{x}_{{m}}^{{2}}}={c}_{{m}}$$ either by finding convenient representation for $$\displaystyle{a}_{{m}}{\quad\text{and}\quad}{b}_{{m}}$$ or by using the inverse of $$\displaystyle{a}_{{m}}$$. Then find a solution for this congruence directly or by replacing $$\displaystyle{c}_{{m}}$$ : with its appropriate representative in $$\displaystyle\frac{{Z}}{{{m}{Z}}}$$. If there is no solution explain why. Here $$\displaystyle{a}_{{m}},{b}_{{m}},{x}_{{m}}{\left(={x}\right)},{c}_{{m}}\in\frac{{Z}}{{{m}{Z}}}:$$
$$\displaystyle{I}{n}\frac{{Z}}{{{19}{Z}}},{\left[{2}\right]}\cdot{x}^{{2}}={\left[{13}\right]}:$$

2021-01-31
Step 1
Let $$\displaystyle{y}={x}^{{2}}$$ then the given congruence relation becomes,
$$\displaystyle{a}{y}\equiv{b}\text{mod}{m}{t}\hat{{i}}{s}{2}{y}\equiv{13}\text{mod}{19}$$
This congruence has unique solution if and only if gcd(a,m)=1
Here gcd(2,19)=1 hence the given congruence has unique solution.
Now since 2 has an inverse, we get $$\displaystyle{y}\equiv{2}^{{−{1}}}{13}\text{mod}{19}$$ which is the only solution.
The inverse of $$\displaystyle{2}\in\mathbb{Z}_{{{19}}}$$ is 10.
$$\displaystyle{y}\equiv{130}\text{mod}{19}\Rightarrow{y}={16}$$
Now $$\displaystyle{y}={x}^{{2}}\Rightarrow{16}={4}^{{2}}$$. Hence x =4 is the required solution.

### Relevant Questions

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
Ok, If a bobsled makes a run down an ice track starting at 150m vertical distance up the hill and there is no friction, what isthe velocity at the bottom of the hill?
I know that the initial velocity here is 0 because it isstarting from rest. And this problem deal with theconservation of energy. But, I don't know where to go fromhere.
The student engineer of a campus radio station wishes to verify the effectivencess of the lightning rod on the antenna mast. The unknown resistance $$\displaystyle{R}_{{x}}$$ is between points C and E. Point E is a "true ground", but is inaccessible for direct measurement because the stratum in which it is located is several meters below Earth's surface. Two identical rods are driven into the ground at A and B, introducing an unknown resistance $$\displaystyle{R}_{{y}}$$. The procedure for finding the unknown resistance $$\displaystyle{R}_{{x}}$$ is as follows. Measure resistance $$\displaystyle{R}_{{1}}$$ between points A and B. Then connect A and B with a heavy conducting wire and measure resistance $$\displaystyle{R}_{{2}}$$ between points A and C.Derive a formula for $$\displaystyle{R}_{{x}}$$ in terms of the observable resistances $$\displaystyle{R}_{{1}}$$ and $$\displaystyle{R}_{{2}}$$. A satisfactory ground resistance would be $$\displaystyle{R}_{{x}}{<}{2.0}$$ Ohms. Is the grounding of the station adequate if measurments give $$\displaystyle{R}_{{1}}={13}{O}{h}{m}{s}$$ and R_2=6.0 Ohms?

For the following statement, either prove that they are true or provide a counterexample:
Let a, b, c, $$\displaystyle{m}\in{Z}$$ such that m > 1. If $$\displaystyle{a}{c}\equiv{b}{c}{\left(\mod\right)},{t}{h}{e}{n}\ {a}\equiv{b}{\left(\mod{m}\right)}$$

Give a full and correct answer Why is it important that a sample be random and representative when conducting hypothesis testing? Representative Sample vs. Random Sample: An Overview Economists and researchers seek to reduce sampling bias to near negligible levels when employing statistical analysis. Three basic characteristics in a sample reduce the chances of sampling bias and allow economists to make more confident inferences about a general population from the results obtained from the sample analysis or study: * Such samples must be representative of the chosen population studied. * They must be randomly chosen, meaning that each member of the larger population has an equal chance of being chosen. * They must be large enough so as not to skew the results. The optimal size of the sample group depends on the precise degree of confidence required for making an inference. Representative sampling and random sampling are two techniques used to help ensure data is free of bias. These sampling techniques are not mutually exclusive and, in fact, they are often used in tandem to reduce the degree of sampling error in an analysis and allow for greater confidence in making statistical inferences from the sample in regard to the larger group. Representative Sample A representative sample is a group or set chosen from a larger statistical population or group of factors or instances that adequately replicates the larger group according to whatever characteristic or quality is under study. A representative sample parallels key variables and characteristics of the large society under examination. Some examples include sex, age, education level, socioeconomic status (SES), or marital status. A larger sample size reduced sampling error and increases the likelihood that the sample accurately reflects the target population. Random Sample A random sample is a group or set chosen from a larger population or group of factors of instances in a random manner that allows for each member of the larger group to have an equal chance of being chosen. A random sample is meant to be an unbiased representation of the larger population. It is considered a fair way to select a sample from a larger population since every member of the population has an equal chance of getting selected. Special Considerations: People collecting samples need to ensure that bias is minimized. Representative sampling is one of the key methods of achieving this because such samples replicate as closely as possible elements of the larger population under study. This alone, however, is not enough to make the sampling bias negligible. Combining the random sampling technique with the representative sampling method reduces bias further because no specific member of the representative population has a greater chance of selection into the sample than any other. Summarize this article in 250 words.
Determine which equations are linear equations in the variables x, y, and z. If any equation is not linear, explain why not.
$$3\cos x-4y+z=\sqrt{3}$$
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(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than $$\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}$$.
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of $$\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}$$? Explain, based on theprobability of this occurring.
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(e) What is the moment generating function for X?
The value of the operation [9]+[8] in $$\displaystyle{Z}_{{{13}}}$$ and to write the answer in the form [r] with $$\displaystyle{0}\leq{r}{<}{m}$$.
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