# is Bivariate normal distributions an assumption for the Pearson Correlation ?

Question
Normal distributions
is Bivariate normal distributions an assumption for the Pearson Correlation ?

2020-11-07
Step 1
If we suspect linear relationship between two variables, then for a given set of data, we can evaluate the value of Pearson correlation coefficient without having any assumption. And we can have a sense of with what intensity the variables are linearly related.
Step 2
But if we would like to test the significance of a given correlation value for a set of observations, and we would like to infer whether the data really gives sufficient evidence in favor of significant correlation or linear relationhip between considered variables, then for the construction of test statistics, assumption of normality is necessary.

### Relevant Questions

Np>5 and nq>5, estimate P(at least 11) with n=13 and p = 0.6 by using the normal distributions as an approximation to the binomial distribution. if $$\displaystyle{n}{p}{<}{5}{\quad\text{or}\quad}{n}{q}{<}{5}$$ then state the normal approximation is not suitable.
p (at least 11) = ?
Decide which of the following statements are true.
-Normal distributions are bell-shaped, but they do not have to be symmetric.
-The line of symmetry for all normal distributions is x = 0.
-On any normal distribution curve, you can find data values more than 5 standard deviations above the mean.
-The x-axis is a horizontal asymptote for all normal distributions.
Select all that apply. We show that our sample statistics have (at minimum) a somewhat normal distribution because
this allows us to use t and z critical values.
this allows us to use t and z tables for probabilities.
this tells us that our sampling is appropriate.
normal distributions are cool and that's all we talk about in this class.
$$\displaystyle{\left(\mu{1}-\mu{2}\right)}$$ For two normal distributions
Obtain the appropriate point estimator
Let two independent random samples, each of size 10, from two normal distributions $$\displaystyle{N}{\left(\mu_{{1}},\sigma_{{2}}\right)}{\quad\text{and}\quad}{N}{\left(\mu_{{2}},\sigma_{{2}}\right)}$$ yield $$\displaystyle{x}={4.8},{{s}_{{1}}^{{2}}}$$
$$\displaystyle={8.64},{y}={5.6},{{s}_{{2}}^{{2}}}$$
= 7.88.
Find a 95% confidence interval for $$\displaystyle\mu_{{1}}−\mu_{{2}}$$.
There is a direct relationship between the chi-square and the standard normal distributions, whereby the square root of each chi-square statistic is mathematically equal to the corresponding z statistic at significance level $$\displaystyle\alpha$$.
1.True
2.False
A fair coin is flipped 104 times. Let x be the number of heads. What normal distributions best approximates x?
Which of the following is(are) True?
I. The means of the Student’s t and standard normal distributions are equal.
II. The standard normal distribution approaches the Student’s t distribution as the degrees of freedom becomes large.
A. I only
B. II only
C. Both I and II
D. Neither I nor II