Scores on a test are normally distributed with a mean of 68.2 and a standard deviation of 10.4. Estimate the probability that among 75 randomly selscted students, at least 20 of them score greater that 78.

Use z-scores.

z = (x - mean)/sd

With your data:

z = (78 - 68.2)/10.4 = 0.94

.1736 is the probability using a z-table for a single student with a score greater than 78.

Now we can use a normal approximation to the binomial distribution.

mean = np = (75)(.1736) = 13
standard deviation = √np(1-p) = √(75)(.1736)(.8264) = 3.28

Again, use z-scores.
z = (20 - 13)/3.28 = 2.13

Use the z-table to find the probability. (Remember the problem says "at least 20" which means 20 or more.)

I hope this will help.

To estimate the probability of at least 20 students scoring greater than 78 out of a sample of 75 students, we can use the z-score formula and the standard normal distribution.

Step 1: Calculate the z-score for the given score of 78.
The z-score formula is given by: z = (x - μ) / σ
Where:
- x is the value we want to convert to a z-score (in this case, 78),
- μ is the population mean (68.2), and
- σ is the population standard deviation (10.4).

z = (78 - 68.2) / 10.4
z ≈ 0.9423

Step 2: Calculate the probability of a randomly selected student scoring greater than 78 using the z-table.
Using a z-table (or z-score calculator), we can find that the probability of a student scoring less than 0.9423 is approximately 0.8273.

Step 3: Calculate the probability of at least 20 out of 75 students scoring greater than 78.
To calculate the probability of at least 20 students scoring greater than 78, we need to find the probability of exactly 20, 21, 22, ..., up to 75 students, and then subtract this cumulative probability from 1.

P(x ≥ 20) = 1 - P(x < 20)
= 1 - P(x ≤ 19)

Step 4: Calculate the z-score for 19 students using the formula from Step 1.
z = (19 - 68.2) / 10.4
z ≈ -4.712

Step 5: Find the probability of 19 students scoring less than 78 using the z-table.
Using the z-table, we find that the probability of a student scoring less than -4.712 is negligible (close to 0).

Step 6: Calculate the probability of at least 20 students scoring greater than 78.
P(x ≥ 20) = 1 - P(x < 20)
= 1 - P(x ≤ 19)
= 1 - 0 (since P(x ≤ 19) is negligible)

Therefore, the estimated probability that among 75 randomly selected students, at least 20 of them score greater than 78 is 1 or 100%.

To estimate the probability of at least 20 students scoring greater than 78 out of a sample of 75 students, we can use the normal distribution and the properties of z-scores.

First, we need to standardize the scores using the formula for z-scores: z = (x - μ) / σ, where x is the individual score, μ is the mean, and σ is the standard deviation.

Let's find the z-score associated with a score of 78:
z = (78 - 68.2) / 10.4
z = 0.9423

Next, we'll find the probability associated with this z-score using a standard normal table or calculator. We want to find P(Z > 0.9423), where Z represents a standard normal random variable. Looking up this value in a standard normal table, we find that the probability is approximately 0.1736.

Now, we need to consider the sample of 75 students. Since the scores are independent, we can use the binomial distribution to model the number of students scoring greater than 78.

Using the binomial distribution equation: P(X ≥ k) = 1 - P(X < k), where X is the number of students scoring greater than 78, and k is the minimum number of students.

We want to find P(X ≥ 20) for our given sample size of 75 students. We can use the binomial probability formula to calculate this:

P(X ≥ 20) = 1 - P(X < 20)
= 1 - sum(C(n, x) * p^x * q^(n-x)) for x = 0 to 19, where n is the sample size, p is the probability of success (probability of scoring greater than 78), and q is the probability of failure (probability of scoring less than or equal to 78).

Using these values:
n = 75, p = 0.1736, q = 1 - p = 0.8264

We calculate:
P(X ≥ 20) = 1 - sum(C(75, x) * (0.1736)^x * (0.8264)^(75-x)) for x = 0 to 19

Calculating this sum may be time-consuming by hand, but you can use statistical software or an online calculator to quickly find the result.