The image contains four statistical problems involving the calculation of quartiles and percentile ranks based on given data sets. Here are the solutions for each problem:

Problem 1:

The data set represents the number of keyboards assembled at a company for 25 days:

Data Set:
45, 52, 48, 49, 51, 56, 45, 41, 46, 50,
44, 42, 52, 50, 46, 51, 54, 47, 44, 48,
53, 48, 49, 52, 49

a) Calculate the value of the third quartile (Q3).

To find Q3:

1. Arrange the data in ascending order.
2. Identify Q3, which is the 75th percentile.

Sorted Data:
41, 42, 44, 44, 45, 45, 46, 46, 47, 48,
48, 48, 49, 49, 49, 50, 50, 51, 51, 52,
52, 52, 53, 54, 56

Position of Q3: $Q3 = \frac{3(n+1)}{4} = \frac{3(25+1)}{4} = 19.5$
Q3 is between the 19th and 20th values, i.e., between 51 and 52.
Q3 Value: $51.5$

b) Find the percentile rank of 50.

Percentile rank = $\frac{\text{number of values below 50} + 0.5}{\text{total number of values}} \times 100$.

Values below 50: 15 values
Percentile Rank = $\frac{15 + 0.5}{25} \times 100 = 62\%$.

Problem 2:

Data Set: 50, 71, 72, 72, 78, 80, 82, 83, 84, 84, 85, 86, 86, 87, 90, 94, 96, 98, 99.

a) Find the percentile rank for a score of 84.

Number of values below 84: 8
Percentile Rank = $\frac{8 + 0.5}{19} \times 100 = 44.74\%$.

Problem 3:

Data Set: 72, 74, 76, 78, 80, 84, 84, 85, 86, 88, 89, 91, 94, 96, 98.

a) Find the percentile rank of 89.

Number of values below 89: 10
Percentile Rank = $\frac{10 + 0.5}{15} \times 100 = 70\%$.

Problem 4:

Data Set: 58, 59, 62, 66, 68, 69, 70, 71, 71, 71, 73.

a) Find the percentile rank for score 51.

The score 51 is not present in the data. Assuming you need to find the rank closest to 51:

Values below 51: 0 (because 58 is the lowest).
Percentile Rank = $0\%$.

Would you like more detailed explanations on these solutions or have any specific questions?

Further Questions:

1. How do you determine the first quartile (Q1) in a data set?
2. What is the significance of the median in a quartile calculation?
3. How can percentile rank be useful in statistical analysis?
4. What is the difference between the 25th and 75th percentiles?
5. How do you calculate the interquartile range (IQR)?

Tip: Quartiles and percentiles help in understanding data distribution, especially in comparing individual scores against the overall set.