Question

Recent research suggests that many American have deficient magnesium levels. Vitamin D, which increases calcium and phosphate levels, cannot be metabolized without magnesium, but this study suggests that many American are consuming only half the recommended daily allowance of magnesium. A random sample of 1200 American was selected, and each was tested for magnesium level. In total, 540 were found to have deficient magnesium levels.

a) Find the value of the point estimate of the true proportion of American who have deficient magnesium levels.
b) Construct a 95% confidence interval for the percentage of all American who have deficient magnesium levels.
c) Interpret the resulting confidence interval in part (b).
d) Suppose the confidence interval obtained in part (b) is too wide. How can the width of this interval be reduced? Discuss all possible alternatives. Which alternative is the best?
e) The American Medical Association claims that less than 50% of American who are magnesium deficient. Can we believe this claim? Use ?=0.02 and the critical value approach.
f) Assume that we want to select another sample to construct a 92% confidence interval for the percentage of all American who have deficient magnesium level with a margin of error of 0.02. Using the above sample as a preliminary study, determine the minimum sample size needed to select the new sample in this case.

Answer 1

a)

b)

The confidence interval for the proportion is obtained using the formula,

Where,

c)

We are 95% confident that the population proportion of Americans who have deficient magnesium levels lies in the range (0.422,0.478).

d)

The formula for the confidence interval is defined as,

The size of the confidence interval depends on the significance level and the sample size hence we can reduce the interval either by increasing the significance level alpha or by increasing the sample size.

The best way is to increase the sample size.

e)

Hypotheses:

The null and alternative hypotheses are defined as,

This is a left-tailed test

the significance level = 0.02

Critical Value:

The critical value for the z statistic is obtained from the standard normal distribution table for significance level = 0.02 for the one-tailed test.

Test statistic:

The z-statistic is obtained using the formula,

Conclusion:

Since the z statistic is greater than the critical value, the null hypothesis is rejected. Hence we can conclude that less than 50% of Americans are magnesium deficient.

f)

The sample size is obtained using the formula,

Where,

Let  

Now,