Adolfo Hebert
2022-03-21
Answered

Let $X}_{1},\dots {X}_{n$ be iid $N(\theta ,\text{}1)$ . A $95\mathrm{\%}$ confidence interval for $\theta$ is $\stackrel{\u2015}{X}\pm \frac{1.96}{\sqrt{n}}$ . Let p denote the probability that an additional independent observation, $X}_{n+1$ , will fall in this interval. Is p greater than, less than, equal to $0.95$ ? Prove your answer.

You can still ask an expert for help

Jaslyn Allison

Answered 2022-03-22
Author has **13** answers

Step 1

Perhaps a little more than a hint but here goes

The question appears to be asking if

$p=P\{\stackrel{\u2015}{X}-1.96\sqrt{\frac{1}{n}}\le {X}_{n+1}\le \stackrel{\u2015}{X}+1.96\sqrt{\frac{1}{n}}\}$

$=P\{-1.96\sqrt{\frac{1}{n}}\le {X}_{n+1}-\stackrel{\u2015}{X}\le 1.96\sqrt{\frac{1}{n}}\}$

is less than, equal to, or greater than 0.95.

Now, it turns out $X}_{n+1}-\stackrel{\u2015}{X$ (itself a linear combination of normal random variables) is also normal, with mean 0 and variance $1+\frac{1}{n}$ (due to independence of $X}_{n+1$ from the other $X}_{i$ and hence from $\stackrel{\u2015}{X}$).

Then we have that

$P\{-1.96\sqrt{1+\frac{1}{n}}\le {X}_{n+1}-\stackrel{\u2015}{X}\le 1.96\sqrt{1+\frac{1}{n}}\}=0.95$ as well.

Now take a look at the interval above and notice that

$[-1.96\sqrt{1+\frac{1}{n}},\text{}\text{}1.96\sqrt{1+\frac{1}{n}}\text{}]$

$=[-1.96\sqrt{1+\frac{1}{n}},\text{}\text{}-1.96\sqrt{\frac{1}{n}}\text{})$

$\bigcup {\begin{array}{cc}-1.96\sqrt{\frac{1}{n}}& \text{}\text{}1.96\sqrt{\frac{1}{n}}\text{}\end{array}}$

$\bigcup (1.96\sqrt{\frac{1}{n}},\text{}\text{}1.96\sqrt{1+\frac{1}{n}}\text{}]$

Can you deduce where p stands relative to $0.95$?

asked 2021-08-03

A simple random sample of 60 items resulted in a sample mean of 80. The population standard deviation is $\sigma =15$

a) Compute the$95\mathrm{\%}$ confidence interval for the population mean. Round your answers to one decimal place.

b) Assume that the same sample mean was obtained from a sample of 120 items. Provide a$95\mathrm{\%}$ confidence interval for the population mean. Round your answers to two decimal places.

c) What is the effect of a larger sample size on the interval estimate?

Larger sample provides a-Select your answer-largersmallerItem 5 margin of error.

a) Compute the

b) Assume that the same sample mean was obtained from a sample of 120 items. Provide a

c) What is the effect of a larger sample size on the interval estimate?

Larger sample provides a-Select your answer-largersmallerItem 5 margin of error.

asked 2021-08-09

A clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjects. Before treatment, 13 subjects had a mean wake time of 101.0 min. After treatment, the 13 subjects had a mean wake time of 94.6 min and a standard deviation of 24.9 min. Assume that the 13 sample values appear to be from a normally distributed population and construct a

Construct the

What does the result suggest about the mean wake time of 101.0 min before the treatment? Does the drug appear to be effective?

The confidence interval ▼ does not include| includes the mean wake time of 101.0 min before the treatment, so the means before and after the treatment ▼ could be the same |are different. This result suggests that the drug treatment ▼ does not have | has a significant effect.

asked 2021-01-10

The average zinc concentration recovered from a sample of measurements taken in 36 different locations in a river is found to be 2.6 grams per liter. Find the 95% confidence intervals for the mean zinc concentration in the river. Assume that the population standard deviation is 0.3 gram per liter.

asked 2021-08-12

In a science fair project, Emily conducted an experiment in which she tested professional touch therapists to see if they could sense her energy field. She flipped a coin to select either her right hand or her left hand, and then she asked the therapists to identify the selected hand by placing their hand just under Emily's hand without seeing it and without touching it. Among 358 trials, the touch therapists were correct 172 times. Complete parts (a) through (d).

a) Given that Emily used a coin toss to select either her right hand or her left hand, what proportion of correct responses would be expected if the touch therapists made random guesses? (Type an integer or a decimal. Do not round.)

b) Using Emily's sample results, what is the best point estimate of the therapists' success rate? (Round to three decimal places as needed.)

c) Using Emily's sample results, construct a

Round to three decimal places as needed - ?

asked 2021-10-26

Consider two continuous random variables X and Y with joint density function

$f(x,y)=beg\in \left\{cases\right\}x+y\text{}o\le x\le 1,0\le y\le 1\mathrm{\setminus}0\text{}\text{}\text{}\text{}otherwiseend\left\{cases\right\}$

P(X>0.8, Y>0.8) is?

P(X>0.8, Y>0.8) is?

asked 2021-10-19

Random variables $X}_{1},{X}_{2},\dots ,{X}_{n$ are independent and identically distributed; 0 is a parameter of their distribution.

If$q(X,0)\sim N(0,1)$ is a pivotal function for 0, explain how you would use this result to obtain a symmetrical 95% confidence interval for 0.

If

asked 2020-11-22

List the assumptions necessary for each of the following inferential techniques:
a. Large-sample inferences about the difference $({\mu}_{1}-{\mu}_{2})$ between population means using a two-sample z-statistic
b. Small-sample inferences about $({\mu}_{1}-{\mu}_{2})$ using an independent samples design and a two-sample t-statistic
c. Small-sample inferences about $({\mu}_{1}-{\mu}_{2})$ using a paired difference design and a single-sample t-statistic to analyze the differences
d. Large-sample inferences about the differences $({\mu}_{1}-{\mu}_{2})$ between binomial proportions using a two sample z-statistic e. Inferences about the ratio $\frac{{\sigma}_{1}^{2}}{{\sigma}_{2}^{2}}$ of two population variances using an F-test.