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 =

To find the probability P(X<40) using the normal approximation and a TI-84 Plus calculator, follow these steps:

1. 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)) = 4.49.

2. Next, you need to standardize the value X=40 to a z-score. The z-score is calculated as (X - μ) / σ. So, z = (40 - 33.54) / 4.49 = 1.44.

3. Now, you can use the TI-84 Plus calculator to find the probability. Go to the distribution menu by pressing 2ND VARS. Choose "2: normalcdf". You will see "normalcdf(" on the screen.

4. Enter the values for the lower limit, upper limit, mean, and standard deviation. In this case, the lower limit is -∞, which we can approximate as -1E99 on the calculator. The upper limit is the z-score 1.44. The mean is 0 and the standard deviation is 1 for the standard normal distribution. So, you should enter -1E99, 1.44, 0, 1.

5. Press ENTER to calculate the probability. The calculator will display the probability, rounded to at least four decimal places.

Please note that the above steps are for finding P(X<40). If you want to find P(X<=40), you should use X=40.5 instead of X=40 when calculating the z-score, because of the continuity correction.

To find the probability P(X<40) using the normal approximation and a TI-84 Plus calculator, follow these steps:

1. 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.

2. Next, we need to standardize the value X=40 to a z-score. The z-score is calculated as (X - μ) / σ. So, z = (40 - 33.54) / 4.47 = 1.44.

3. Now, we want to find P(X<40), which is the same as finding P(Z<1.44) in the standard normal distribution.

4. To find this probability using a TI-84 Plus calculator, press 2ND, then VARS to get to the distribution menu. Select "2: normalcdf" to get to the normal cumulative distribution function.

5. Enter the lower limit, upper limit, mean, and standard deviation. Here, the lower limit is -∞, which we can approximate as -1E99 on the calculator. The upper limit is the z-score 1.44. The mean is 0 and the standard deviation is 1 for the standard normal distribution.

6. So, you should enter: normalcdf(-1E99, 1.44, 0, 1). Press ENTER.

7. The calculator will give you the probability, rounded to at least four decimal places.

Please note that the TI-84 Plus calculator gives the probability for a continuous approximation of the binomial distribution, so the result might be slightly different from the exact binomial probability.