Hi need help with these problems from my statistics homework. We use r studio for assignments. Can I please get help with these questions. Can you answer and explain how you got to the answer please?

Problem 1:

Researchers investigating characteristics of gifted children collected data from schools in a large city on a random sample of thirty-six children who were identified as gifted children soon after they reached the age of four.

In this study, along with variables on the children, the researchers also collected data on the mother's and father's IQ of the $n=36$ randomly sampled gifted children. The histogram below shows the distribution of mother's IQ.Also provided are some sample statistics.

```{r}

data(gifted)

# compute number of observations:

with(gifted,length(motheriq))

with(gifted,mean(motheriq))

with(gifted,sd(motheriq))

with(gifted,hist(motheriq))

t.test(gifted$motheriq,mu=100)

```

Perform a hypothesis test to evaluate if these data provide convincing evidence that the average IQ of mothers of gifted children is different than the average IQ for the population at large, which is 100. Use the traditional p-value of 0.05 or less to indicate evidence to doubt the null hypothesis.

* Hypotheses :$H_{0}: \mu = 100$versus $H_{a}: \mu eq 100$.

* Test Statistic Value:

* P-Value:

* Conclusion:

*(part 1)

*(part 2)

*(part 3)

*(part 4)

#### Problem 1: part b

Calculate a 90 percent confidence interval for the average IQ of mothers of gifted children.

Remember the formula for this method is $\bar{x} \pm T^{*} (S/\sqrt{n})$.

```{r}

# you can have the t.test compute the interval for you if you wish...

#t.test(gifted$motheriq,conf.level=.90)

```

Answer:

(part c)Use the confidence interval boundaries produced in part (b) to make a conclusion about the population mean IQ for mothers of gifted children.Does this population mean differ from 100 ? Give a full argument using your previous work.

Answer:

(d) What sample size is needed to obtain a margin of error of 2.5 IQ points in a 95 percent confidence interval for the population mean?

Answer: