Below, n is the sample size, p is the population proportion of successes, and X is the number of successes in the sample. Use the normal approximation and the TI-84 Plus calculator to find the probability. Round the answer to at least four decimal places.=n=78, p0.43P<X40
Question
Below, n is the sample size, p is the population proportion of successes, and X is the number of successes in the sample. Use the normal approximation and the TI-84 Plus calculator to find the probability. Round the answer to at least four decimal places.=n=78, p0.43P<X40
Solution
To find the probability P(X<40) using the normal approximation and a TI-84 Plus calculator, follow these steps:
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First, calculate the mean (μ) and standard deviation (σ) of the distribution. The mean is np and the standard deviation is sqrt(np(1-p)). Here, n=78 and p=0.43. So, μ = np = 780.43 = 33.54 and σ = sqrt(np(1-p)) = sqrt(780.43*(1-0.43)) = 4.47.
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Next, standardize the value X=40 using the formula Z = (X - μ) / σ. So, Z = (40 - 33.54) / 4.47 = 1.44.
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Now, use the TI-84 Plus calculator to find the probability. Press 2ND, then VARS to get to the distribution menu. Choose 2:normalcdf. Then enter the values: lower limit is -9999 (a very large negative number to represent negative infinity), upper limit is 1.44 (the Z value we calculated), mean is 0, and standard deviation is 1. Press ENTER to calculate the probability.
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The calculator will give you the probability P(Z<1.44), which is the same as P(X<40). Round this to at least four decimal places.
Please note that the normal approximation is only valid if np and n(1-p) are both greater than 5, which is true in this case.
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