Vitamin D and IBS ~A recent study from the University of Sheffield in the United Kingdom revealed that in a small pilot study of patients with Irritable Bowel Syndrome (IBS), 82% had a vitamin D deficiency.Simon and Vicky read about this pilot study and want to investigate it among patients suffering from IBS in the Northeastern region of the United States. They recruit a random sample of 100 IBS patients in this region and use blood tests to determine that 70 of these patients are vitamin D deficient.Round all calculated answers to 4 decimal places where appropriate.1.Using the information from the sample taken by Simon and Vicky, construct a 99% confidence interval for the true proportion of IBS patients in the Northeastern region of the US who are vitamin D deficient.(,)

2.Which of the following conditions must be met for the confidence interval to be valid? Select all that apply.

A.There must be at least 10 'success' and 10 'failure' observations in the sample.

B.The sample proportion must be normally distributed.

C.The observations must be independent of one another.

D.The value for p must be less than 0.10 to provide evidence against the null hypothesis.

3.The information from the sample collected by Simon and Vicky gives a standard error of

=

SEp^=0.0458. Which of the statements below is a correct interpretation of the standard error?A. On average, for repeated samples of this size, we expect the sample proportion to be approximately 0.0458 from the true population proportion.

B. We can be 4.58% confident that our sample proportion is correctly calculated.

C. The sample proportion is different from the true proportion of IBS patients with vitamin D deficiency approximately 4.58% of the time.

D. We have strong evidence that the true proportion of IBS patients with vitamin D deficiency is contained in a confidence interval.

4.If Simon and Vicky want to estimate the true proportion of IBS patients in the Northeastern area who are vitamin D deficient with 99% confidence and a margin of error of no more than 8.5%, what size sample should they take?Use the value from the University of Sheffield study as a planning value for

p, that is,

=0.82

p=0.82. Round your

zvalue to exactly 3 decimal places.=

n=

5.Researchers in the Southwest region of the United States also replicate this study and calculate a 95% confidence interval for the proportion of IBS patients with vitamin D deficiency in this region to be (0.65, 0.74).Which of the following statements are appropriate interpretations in this scenario? Select all that apply.

A.We can be 95% confident that, on average, the margin of error will vary no more than the size of the standard error.

B.If the researchers collected 100 samples of size n = 100 from this population and constructed 100 confidence intervals, they could expect approximately 95 of them to contain the true proportion of IBS patients with vitamin D deficiency in this region.

C.We can be 95% confident that the true proportion of IBS patients with vitamin D deficiency in this region to be contained in the interval (0.65, 0.74).

D.On average, 95% of the time we can expect any sample proportion from a sample of 100 IBS patients with vitamin D deficiency to be in the interval (0.65, 0.74).