Question

1. shannon is traveling from nyc to d.c. she wants to go by train so she can see the views. since she will be driving home with a family member, she only prices one - way tickets for any time of day on february 15th. below is an ordered listing of all fares that were available for selection on that day. chart with numbers

a. find the percentile rank for a fare of $119
b. find the percentile rank for a fare of $272.
ur class average height is __ inches with a standard deviation of __ in, find you re with respect to the average.
k about it: what does a positive z - score mean?

Explanation:

Part a: Percentile Rank for $119$

Step 1: Count total data points

There are 6 rows and 15 columns, so total data points \( n = 6 \times 15 = 90 \).

Step 2: Count data points less than 119

Looking at the data, the values less than 119: Let's check each row. The first few values: 49, 49, 88, 88, 88, 88, 88, 88 (wait, let's count properly). Wait, the first row: 49, 49, 88, 88, 118, ... So values less than 119: Let's count how many are less than 119. Let's list the counts:

First, the values less than 119: Let's see, the numbers before 119 (or equal? No, percentile rank formula: number of values less than \( x \) divided by total, times 100. Wait, formula for percentile rank: \( \text{Percentile Rank} = \frac{\text{Number of values less than } x}{\text{Total number of values}} \times 100 \)

Wait, let's count the number of values less than 119. Let's look at each row:

Row 1: 49, 49, 88, 88, 118, 133, 133, 133, 161, 171, 173, 173, 173, 272, 284 – values less than 119: 49,49,88,88,118? Wait 118 is less than 119? Yes. Wait, 118 < 119. Then row 1: 5 values (49,49,88,88,118)? Wait no, 49,49,88,88,118 – that's 5? Wait 49 (2), 88 (2), 118 (1) – total 5? Wait row 1 has 15 elements. Let's check row 2: 49, 49, 88, 89, 119, ... Wait 119 is not less than 119. So row 2: 49,49,88,89 – 4 values less than 119.

Row 3: 88, 88, 88, 119, ... Wait 119 is not less than 119. So row 3: 88,88,88 – 3 values? Wait no, 88,88,88 – 3? Wait row 3: first three 88s, then 119. So 3 values less than 119?

Wait maybe I made a mistake. Let's count all values less than 119:

Looking at the data:

First, list all values:

Row 1: 49, 49, 88, 88, 118, 133, 133, 133, 161, 171, 173, 173, 173, 272, 284

Row 2: 49, 49, 88, 89, 119, 133, 133, 161, 161, 173, 173, 173, 173, 272, 284

Row 3: 88, 88, 88, 119, 119, 133, 133, 161, 161, 173, 173, 173, 272, 272, 284

Row 4: 49, 49, 88, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 272, 284

Row 5: 89, 89, 89, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 284, 284

Row 6: 89, 89, 89, 119, 133, 133, 133, 161, 171, 173, 173, 173, 272, 284, 284

Now, count values less than 119:

Row 1: 49,49,88,88,118 – 5 values (118 < 119)

Row 2: 49,49,88,89 – 4 values (119 is not less than 119)

Row 3: 88,88,88 – 3 values (119 is not less than 119)

Row 4: 49,49,88 – 3 values (119 is not less than 119)

Row 5: 89,89,89 – 3 values (119 is not less than 119)

Row 6: 89,89,89 – 3 values (119 is not less than 119)

Wait, that can't be right. Wait row 1: 49,49,88,88,118 – 5. Row 2: 49,49,88,89 – 4. Row 3: 88,88,88 – 3. Row 4: 49,49,88 – 3. Row 5: 89,89,89 – 3. Row 6: 89,89,89 – 3. Wait total less than 119: 5 + 4 + 3 + 3 + 3 + 3 = 21? Wait no, maybe I miscounted.

Wait another approach: count the number of values less than 119. Let's list all values and count:

First, the numbers less than 119: 49, 49, 88, 88, 118, 49, 49, 88, 89, 88, 88, 88, 49, 49, 88, 89, 89, 89, 89, 89, 89. Wait this is getting confusing. Wait maybe the total number of data points is 6 rows ×15 columns=90.

Wait let's count the number of 49s: row1:2, row2:2, row4:2 – total 6.

Number of 88s: row1:2, row2:1, row3:3, row4:1 – total 2+1+3+1=7? Wait no, row1: 49,49,88,88,118,... so 2 eights. Row2: 49,49,88,89,... so 1 eight. Row3: 88,88,88,... so 3 eights. Row4: 49,49,88,... so 1 eight. So total 2+1+3+1=7 eights.

Number of 89s: row2:1, row5:3, row6:3 – total 1+3+3=7.

Number of 118: row1:1.

So values less than 119: 49s (6) + 88s (7) + 89s (7) + 118 (1) = 6+7+7+1=21.

Wait 6 (49s) +7 (88s)=13, +7 (89s)=20, +1 (118)=21. Yes, 21 values less than 119.

Step 3: Calculate percentile rank

Percentile Rank = \( \frac{…

Step 1: Count total data points

Same as before, \( n = 90 \).

Step 2: Count data points less than 272

We need to count how many values are less than 272. Let's see the data: the values less than 272 are all the values except those equal to or greater than 272.

Looking at the data, the values 272 and 284. Let's count the number of 272s and 284s, then subtract from total.

First, count 272s:

Row1:1, row2:1, row3:2, row4:2, row5:1, row6:1 – total 1+1+2+2+1+1=8.

Count 284s:

Row1:1, row2:1, row3:1, row4:1, row5:2, row6:2 – total 1+1+1+1+2+2=8.

Wait, but values equal to 272: do we count them as less than? No, percentile rank is number of values less than \( x \), so values less than 272 are total minus (number of 272s + number of 284s). Wait no: 272 is the value, so values less than 272 are all values <272. Let's check the data:

The values are up to 173, then 272, 284. So values less than 272: all values from 49 up to 173.

Let's count the number of values less than 272. Total data points:90. Number of values ≥272: number of 272s + number of 284s.

From above, 272s:8, 284s:8. Wait but let's check the rows:

Row1: 272 (1), 284 (1) – 2

Row2: 272 (1), 284 (1) – 2

Row3: 272 (2), 284 (1) – 3

Row4: 272 (2), 284 (1) – 3

Row5: 272 (1), 284 (2) – 3

Row6: 272 (1), 284 (2) – 3

Wait total ≥272: 2+2+3+3+3+3=16. Wait 8+8=16? Yes, 8 272s and 8 284s? Wait no, 272s: row1:1, row2:1, row3:2, row4:2, row5:1, row6:1 – total 8. 284s: row1:1, row2:1, row3:1, row4:1, row5:2, row6:2 – total 8. So 8+8=16 values ≥272. Therefore, values less than 272: 90 - 16 = 74? Wait no, wait 272 is the value, so values less than 272 are 90 - (number of 272s + number of 284s). Wait no: if a value is 272, it's not less than 272. So number of values less than 272 is total - number of values ≥272.

Wait let's count the number of 272s:

Row1: 272 – 1

Row2: 272 – 1

Row3: 272, 272 – 2

Row4: 272, 272 – 2

Row5: 272 – 1

Row6: 272 – 1

Total 272s: 1+1+2+2+1+1=8.

Number of 284s:

Row1: 284 – 1

Row2: 284 – 1

Row3: 284 – 1

Row4: 284 – 1

Row5: 284, 284 – 2

Row6: 284, 284 – 2

Total 284s:1+1+1+1+2+2=8.

So total values ≥272:8 (272s) +8 (284s)=16. Therefore, values less than 272:90 -16=74? Wait no, wait 272 is included in ≥272, so values less than 272 are 90 - (number of 272s + number of 284s) =90 -16=74? Wait but let's check with another approach.

Wait the values less than 272: all the values from 49,49,... up to 173,173,173. Let's count the number of values before 272. Each row has 15 elements. Let's see how many elements are before 272 in each row:

Row1: first 13 elements (up to 173,173,173) then 272,284 – so 13 elements less than 272.

Row2: first 13 elements (up to 173,173,173,173) then 272,284 – so 13 elements less than 272.

Row3: first 12 elements (up to 173,173,173) then 272,272,284 – so 12 elements less than 272.

Row4: first 12 elements (up to 173,173,173) then 272,272,284 – so 12 elements less than 272.

Row5: first 12 elements (up to 173,173,173) then 272,284,284 – so 12 elements less than 272.

Row6: first 12 elements (up to 173,173,173) then 272,284,284 – so 12 elements less than 272.

Now sum these:

Row1:13, Row2:13, Row3:12, Row4:12, Row5:12, Row6:12.

Total:13+13=26; 12+12+12+12=48; total 26+48=74. Yes, so 74 values less than 272.

Step 3: Calculate percentile rank

Percentile Rank = \( \frac{74}{90} \times 100 \approx 82.22\% \)

Answer:

(a): Approximately 23.33%

Part b: Percentile Rank for $272$