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Alla [95]
3 years ago
11

From a survey of coworkers you find that 36​% of 200 have already received this​ year's flu vaccine. An approximate 95​% confide

nce interval is ​(0.293​, 0.427​). ​a) How would the confidence interval change if the sample size had been 1800 instead of 200​? ​b) How would the confidence interval change if the confidence level had been 90​% instead of 95​%? ​c) How would the confidence interval change if the confidence level had been 98​% instead of 95​%?
Business
1 answer:
sveta [45]3 years ago
7 0

Answer:

Part a: <em>By increasing the sample size for same confidence level, the confidence interval is reduced.</em>

Part b: <em>By reducing the confidence level for same sample size, the confidence interval is reduced.</em>

Part c: <em>By increasing the confidence level for same sample size, the confidence interval is increased.</em>

Explanation:

Part a

As from given data

  • Percentage of co-workers already received flu vaccine p=0.36,
  • n=1800

The confidence interval is given as

CI=\hat{p} \pm z_{\alpha}\sqrt{\frac{p(1-p)}{n}}

Here

  • z is given for 95% confidence level as 1.960

By substituting values in the equation

CI=\hat{p} \pm z_{\alpha}\sqrt{\frac{p(1-p)}{n}}\\CI=0.36 \pm 1.960\sqrt{\frac{0.36(1-0.36)}{1800}}\\CI=0.36 \pm 0.02217\\CI=(0.3378,0.3812)

It is evident that the confidence interval is smaller, indicating a lesser chance of error.

<em>This means that by increasing the sample size for same confidence level, the confidence interval is reduced.</em>

Part b

As from given data

  • Percentage of co-workers already received flu vaccine p=0.36,
  • n=200

The confidence interval is given as

CI=\hat{p} \pm z_{\alpha}\sqrt{\frac{p(1-p)}{n}}

Here

  • z is given for 90% confidence level as 1.645  

By substituting values in the equation

CI=\hat{p} \pm z_{\alpha}\sqrt{\frac{p(1-p)}{n}}\\CI=0.36 \pm 1.645\sqrt{\frac{0.36(1-0.36)}{200}}\\CI=0.36 \pm 0.0557\\CI=(0.3043,0.4157)

It is evident that the confidence interval is smaller, as the confidence level is reduced.

<em>This means that by reducing the confidence level for same sample size, the confidence interval is reduced.</em>

Part c

As from given data

  • Percentage of co-workers already received flu vaccine p=0.36,
  • n=200

The confidence interval is given as

CI=\hat{p} \pm z_{\alpha}\sqrt{\frac{p(1-p)}{n}}

Here

  • z is given for 98% confidence level as 2.33

By substituting values in the equation

CI=\hat{p} \pm z_{\alpha}\sqrt{\frac{p(1-p)}{n}}\\CI=0.36 \pm 2.33\sqrt{\frac{0.36(1-0.36)}{200}}\\CI=0.36 \pm 0.07899\\CI=(0.2810,0.4389)

It is evident that the confidence interval is greater, as the confidence level is increased.

<em>This means that by increasing the confidence level for same sample size, the confidence interval is increased.</em>

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