Question

The high price of medicines is a source of major expense for those seniors in the United States who have to pay for these medicines themselves. A random sample of 2000 seniors who pay for their medicines showed that they spent an average of $4600 last year on medicines with a standard deviation of$800. Make a 98\% confidence interval for the corresponding population mean.

Confidence intervals
The high price of medicines is a source of major expense for those seniors in the United States who have to pay for these medicines themselves. A random sample of 2000 seniors who pay for their medicines showed that they spent an average of $$\displaystyle\{4600}$$ last year on medicines with a standard deviation of $$\displaystyle\{800}$$. a) Make a $$\displaystyle{98}\%$$ confidence interval for the corresponding population mean. b) Suppose the confidence interval obtained in part a is too wide. How can the width of this interval be reduced? Discuss allpossible alternatives. Which alternative is the best?

2021-07-30
a) First note that it is a one sample test and the population standard deviation is not given. Hence t-distribution will be appropriate in this scenario.
Sample mean, $$\displaystyle\overline{{{x}}}={4600}$$
Sample standard deviation $$\displaystyle{s}={800}$$
Number of samples, $$\displaystyle{n}={2000}$$
Hence, $$\displaystyle{100}{\left({1}-\alpha\right)}\%$$ Confidence interval is given by,
$$\displaystyle{\left(\overline{{{x}}}-{\frac{{{s}}}{{\sqrt{{{n}}}}}}{t}_{{{n}-{1},\ \frac{\alpha}{{2}}}},\ \overline{{{x}}}+{\frac{{{s}}}{{\sqrt{{{n}}}}}}{t}_{{{n}-{1},\ \frac{\alpha}{{2}}}}\right)}$$
Here, $$\displaystyle\alpha={0.01}$$
Hence, Confidence interval is given by,
$$\displaystyle{\left({4558.352},\ {4641.648}\right)}$$
b) Now from the equation of confidence interval, $$\displaystyle{\left(\overline{{{x}}}-{\frac{{{s}}}{{\sqrt{{{n}}}}}}{t}_{{{n}-{1},\ \frac{\alpha}{{2}}}},\ \overline{{{x}}}+{\frac{{{s}}}{{\sqrt{{{n}}}}}}{t}_{{{n}-{1},\ \frac{\alpha}{{2}}}}\right)}$$ we can see that the range will decrease if n increases or the t value decreases. t value will decrease when we decreases $$\displaystyle\alpha$$ keeping the n as fixed. Or increase n when α is fixed. The best way to decrease range is to increase the sample size.