Sharon draws a random card, from a

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.

Random-digit-dialing telephone surveys used to exclude cell phone numbers. If the opinions of people who have only cell phones differed from those of people who have landline service, the poll results may not represent the entire adult population. The Pew Research Center interviewed separate random samples of cell-only and landline telephone users who were less than 30 years old and asked them to describe their political party affiliation.
$\square$ or goodness of fit, homogeneity or independence?

Find the first five terms of the sequence defined by each of these recurrence relations and initial conditions

an = nan−1 + n2an−2, a0 = 1, a1 = 1

 an = an−1 + an−3, a0 = 1, a1 = 2, a2 = 0

Find a vector function that represents the curve of intersection of the two surfaces of the cylinder x^2+y^2=4 and the surface z=xy.

he population of a culture of bacteria is modeled by the logistic equation    

.

To the nearest tenth, how many days will it take the culture to reach 75% of its carrying capacity? What is the carrying capacity? What is the initial population for the model? Why a model like , where is the initial population, would not be plausible? What are the virtues of the logistic model?

For what value of c, is the function continuous on (−∞, ∞)

f(x)={𝑐𝑥^2+2𝑥 𝑖𝑓 𝑥<2
           𝑥^3−𝑐𝑥 𝑖𝑓 𝑥 ≥2