The marks of a large number of students were sampled and the mean and standard deviation were calculated as 42% and 8% respectively.
Assuming this data is normally distributed, what percentage of students scored less than 34%?



Answer :

Answer:

Step-by-step explanation:

We can use the normal distribution table (also called the Z-table) to find the percentage of students who scored less than 34% in this scenario. Here's how:

Standardize the score:

We need to calculate the Z-score, which represents how many standard deviations a particular score (34%) is away from the mean (42%).

Z-score = (X - μ) / σ

where:

X - Individual score (34%)

μ - Mean (42%)

σ - Standard deviation (8%)

Z-score = (34% - 42%) / 8% = -1.00 (round to two decimal places)

Find the percentage using the Z-table:

Look up the Z-score -1.00 in a standard normal distribution table. The table will provide the area (probability) less than that Z-score.

Note: Most Z-tables only show positive Z-scores and the corresponding area. Since the normal distribution is symmetrical, the area less than a negative Z-score is equal to the area greater than its positive counterpart. In this case, the area less than -1.00 is the same as the area greater than 1.00.

Interpret the table value:

Look up 1.00 (or the closest value in the table) in the "greater than" section of the Z-table. This value represents the portion of the population that scored higher than 34%.

For example, if the table value for 1.00 is 0.8413, this means 84.13% of the students scored higher than 34%.

Calculate the percentage scoring less than 34%:

Since the normal distribution is symmetrical, the area less than the mean (42%) is equal to the area greater than the mean. Therefore, to find the percentage scoring less than 34%, subtract the area greater than 34% (obtained from the Z-table) from 1 (which represents the total area under the curve).

Percentage less than 34% = 1 - Area greater than 34% (from Z-table)

For example, if the area greater than 34% was 0.8413 (from step 3), then:

Percentage less than 34% = 1 - 0.8413 = 0.1587

Convert the decimal to percentage (optional):

Multiply the decimal value by 100% to express it as a percentage.

Percentage less than 34% ≈ 15.87% (round to two decimal places)

In conclusion, assuming a normal distribution, approximately 15.87% of the students scored less than 34% in the exam.