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You intend to estimate a population proportion with a confidence interval. The data suggests that the normal distribution is a reasonable approximation for the binomial distribution in this case.Find the critical value that corresponds to a confidence level of 90%. (Report answer accurate to three decimal places with appropriate rounding.)

Question

You intend to estimate a population proportion with a confidence interval. The data suggests that the normal distribution is a reasonable approximation for the binomial distribution in this case.

Find the critical value that corresponds to a confidence level of 90%.
(Report answer accurate to three decimal places with appropriate rounding.)

check_circleAnswer
Step 1

If X is a binomial random variable, then X~B (n, p) where n is the number of trials and p is the probability of success. The proportion is obtained as the fraction of X, the random variable for the number of success with the number of trials (n) or the sample size.

The proportion is denoted by, P^(read as P-hat). That is,

P^=X/n.

The normal distribution can be used to approximate the binomial, when n is large and p is not close to zero or one.

That is, X~N (np, sqrt(npq)).

If the random variable, the mean and the standard deviation is divided with n, a normal distribution of proportions with P^ (called estimated proportion) as the random variable is obtained. That is,

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Step 2

The approximation can be used only if the number of success, np^ and the number of failures, n(1–p^) are both greater than 5.

Step 3

Critical value:

The confidence level is 90%.

Thus, the level of significance is α = 0.10.

Critical value for left tailed test:

The z-critical value for a left tailed test at α = 0.10 is z0.10(=zα).

The corresponding value is obtained using the EXCEL formula,

 “=NORM.INV(0.10,0,1)”.

Thus, the critical value for a left tailed test at 90% confidence level is –1.280.

Critical value for right tailed test:

The z-critical value for a right tailed test at α = 0.10 is z0.90(=z1-α).

The corresponding value is obtained using the EXCEL formula,

“=NORM.INV(0.90,0,1)”.

T...

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