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assume that the helium porosity level (measured in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation 0.75. how large a sample size is necessary if the width of the 95% interval for the true average porosity level is to be 0.40

Sagot :

A large sample size of 15 is necessary if the width of the 95% interval is to be 0.40

Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample. In practice, the sample size used in a study is usually determined based on the cost, time, or convenience of collecting the data, and the need for it to offer sufficient statistical power. In complicated studies there may be several different sample sizes.

Sample sizes may be chosen in several ways:

  1. Using experience – small samples, though sometimes unavoidable, can result in wide confidence intervals and risk of errors in statistical hypothesis testing.
  2. Using a target variance for an estimate to be derived from the sample eventually obtained, i.e., if a high precision is required this translates to a low target variance of the estimator.
  3. Using a target for the power of a statistical test to be applied once the sample is collected.
  4. Using a confidence level, i.e. the larger the required confidence level, the larger the sample size.

Given in the question:

A sample size of n is needed.

n is found when M = 0.4.

95% Central interval

Z = 1.96

  M = z × σ /√n

⇒ 0.40 = 1.96 × 0.75 /√n

⇒ 0.40√n = 1.96 ×0.75

⇒ √n = 1.96 ×0.75/0.40

⇒ (√n)² = (1.96 ×0.75 / 0.40 )²

⇒ n = 14.6

Rounding up 14.6

n = 15

The sample size of 15 is needed.

Thus, A large sample size of 15 is necessary if the width of the 95% interval is to be 0.40

Learn more about Sample Size:

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