Improve Your Understanding of College Level Statistics Problems

Solution verification for hypothesis testing and confidence interval problem
There's a random sample of size ${\displaystyle n=10}$ observations over a normally distributed population with standard deviation ${\displaystyle \sigma =2}$:
8.7, 7, 4, 7.6, 3, 8.1, 6.4, 6.1, 9.4, 6.2
- Find 96% confidence interval for the population mean
- Test the hypothesis ${\displaystyle H_0:\mu =7.5}$ against Ha: ${\displaystyle \mu \ne 7.5}$ with significance level ${\displaystyle \alpha =0.01}$. Find approximation of the observed p value.
My Solution:
We are looking for real numbers l and u s.t.:
${\displaystyle P\left(l<\hat{\theta }<u\right)=0.96}$, where ${\displaystyle \hat{\theta }=\frac{\sum _{i=1}^n{X}_i-n\theta }{\sqrt{n{\sigma }^2}}\sim N(0,1)}$
We are looking for ${\displaystyle z_{0.2}}$ which from the z-score table is: 2.055. Therefore:
$-2.055<\frac{{\sum }_{i=1}^n{X}_i-n\theta }{\sqrt{n{\sigma }^2}}<2.055\equiv -2.055\sqrt{40}<\sum _{i=1}^n{X}_i-n\theta <2.055\sqrt{40}\equiv$

$\equiv ­12.996-66.5<-10\theta <12.996-66.5\equiv 5.350<\theta <7.94.$
For the second part:
We set two z-scores: ${\displaystyle z_1=-2.325\text{and}{z}_2=2.325}$ from which we construct a two-tailed test.
Our test statistic is: ${\displaystyle t=\frac{(6.65-7.5)\sqrt{10}}{2}=-1.343\Rightarrow t\ge -2.235}$ from where we see the test statistic t is not in the rejection region, therefore we fail to reject the null hypothesis.

J. P. Morgan Asset Management publishes information about financial investments. Between 2002 and 2011 the expected return for the S&P was with a standard deviation of and the expected return over that same period for a Core Bonds fund was with a standard deviation of (J. P. Morgan Asset Management, Guide to the Markets). The publication also reported that the correlation between the S&P and Core Bonds is . You are considering portfolio investments that are composed of an S&P index fund and a Core Bonds fund. a.  Using the information provided, determine the covariance between the S&P and Core Bonds. Round your answer to two decimal places. If required enter negative values as negative numbers.

A thousand people answered a questionnaire about a recent primary election with two candidates, Washington and Jefferson. 410 said that they would find Washington to be an acceptable choice, 490 said that they would find Jefferson to be an acceptable choice, and of these 290 said that either Washington or Jefferson would be acceptable.