Sharon draws a random card, from a regular deck of cards, and rolls a regular 6 sided die. What is the probability that Sharon draws a card that is a Heart and rolls a 3?

2022-04-28

Sharon draws a random card, from a regular deck of cards, and rolls a regular 6 sided die. What is the probability that Sharon draws a card that is a Heart and rolls a 3?

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asked 2022-04-19

Solution verification for hypothesis testing and confidence interval problem

There's a random sample of size $n=10$ observations over a normally distributed population with standard deviation $\sigma =2$:

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 ${H}_{0}:\mu =7.5$ against Ha: $\mu \ne 7.5$ with significance level $\alpha =0.01$. Find approximation of the observed p value.

My Solution:

We are looking for real numbers l and u s.t.:

$P\left(l<\hat{\theta}<u\right)=0.96$, where $\hat{\theta}=\frac{\sum _{i=1}^{n}{X}_{i}-n\theta}{\sqrt{n{\sigma}^{2}}}\sim N(0,1)$

We are looking for $z}_{0.2$ 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 $

$\equiv \xad12.996-66.5<-10\theta <12.996-66.5\equiv 5.350<\theta <7.94.$

For the second part:

We set two z-scores: ${z}_{1}=-2.325\text{}\text{and}\text{}{z}_{2}=2.325$ from which we construct a two-tailed test.

Our test statistic is: $t=\frac{(6.65-7.5)\sqrt{10}}{2}=-1.343\Rightarrow t\ge -2.235$ from where we see the test statistic t is not in the rejection region, therefore we fail to reject the null hypothesis.

asked 2022-04-01

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?

asked 2022-03-27

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

asked 2022-03-29

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.

asked 2022-05-05

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

asked 2022-03-25

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

f(x)={𝑐𝑥^2+2𝑥 𝑖𝑓 𝑥<2

𝑥^3−𝑐𝑥 𝑖𝑓 𝑥 ≥2

asked 2022-03-28