The scores for standardized test are normally distributed with a mean of 300 and standard deviation of 39. If the test is given to 700 students, how many are expected to have scores between 300 and 378.
z = (300-300)/(39/sqrt(700))
z = (378-300)/(39/sqrt(700))
To find the number of students expected to have scores between 300 and 378, we need to calculate the probability of a score falling within this range. Since the scores are normally distributed, we can use the z-score formula to standardize the scores and then look up the corresponding probabilities from a z-table.
The z-score formula is:
z = (x - μ) / σ
Where:
z is the standardized score
x is the individual score
μ is the mean of the distribution
σ is the standard deviation of the distribution
For the lower bound of 300, the z-score is:
z1 = (300 - 300) / 39 = 0
For the upper bound of 378, the z-score is:
z2 = (378 - 300) / 39 = 2
Now, we need to find the probability of getting a z-score between 0 and 2. We can use a standard normal distribution table (z-table) to find this probability.
The table provides areas under the curve for different z-scores. We need to find the area between z = 0 and z = 2. From the table, we find that the area to the left of z = 2 is 0.9772, and the area to the left of z = 0 is 0.5.
To find the area between z = 0 and z = 2, we subtract the area to the left of z = 0 from the area to the left of z = 2:
0.9772 - 0.5 = 0.4772
This represents the probability that a randomly selected student scores between 300 and 378.
To find the number of students expected to fall within this range, we multiply the probability by the total number of students:
Number of students = Probability x Total number of students
Number of students = 0.4772 x 700
Thus, the number of students expected to have scores between 300 and 378 is approximately 334 (rounded to the nearest whole number).