Question

1. in a recent year, consumer reports rated many tablet computers for overall performance and quality. based on several variables, they gave each tablet an overall rating out of 100, where higher scores indicate better tablets. the overall ratings of the 20 tablets produced by apple are given in the chart below, followed by summary statistics.

apple 87 87 87 87 86 86 86 86 84 84 84 84 83 83 83 83 81 79 76 73

apple: (\bar{x} = 83.45), (s_{x}=3.762), min = 73, (q_{1}=83), median = 84, (q_{3}=86), max = 87, iqr = 3

a) the ipad had a score of 81. calculate the percentile and interpret it for the ipad. el ipad obtuvo una puntuación de 81. calculo el percentil e intérpretalo para el ipad (2 pts)

b) interpret the standard deviation of the ratings for apple devices in context. interpretar la desviación estándar de las calificaciones de los dispositivos apple en contexto. (1 pt)

c) is the rating of 76 an outlier? explain. ¿es la calificación de 76 un valor atípico? explícalo. (2 pts)

Explanation:

Part a)

Step1: Count the number of data points less than 81

Looking at the data: 73,76,79,81,83,83,83,83,84,84,84,84,86,86,86,86,87,87,87,87.
The number of data points less than 81 is 3 (73,76,79).

Step2: Calculate the percentile

The formula for percentile \(P=\frac{\text{Number of data points less than }x}{\text{Total number of data points}}\times100\)
Here \(x = 81\), total number of data points \(n=20\)
\(P=\frac{3}{20}\times 100=15\)

Interpretation: The iPad's score of 81 is at the 15th percentile. This means that 15% of the Apple tablets in the sample have an overall rating less than 81.

The standard deviation \(s_x = 3.762\). In the context of tablet ratings (out of 100), the standard deviation measures the typical amount by which the ratings of Apple tablets deviate from the mean rating (\(\bar{x}=83.45\)). A standard deviation of approximately 3.76 indicates that, on average, the ratings of Apple tablets are about 3.76 units away from the mean rating of 83.45.

Step1: Find the lower and upper fences

The inter - quartile range \(IQR = 3\), \(Q_1=83\), \(Q_3 = 86\)
Lower fence \(=Q_1-1.5\times IQR=83-1.5\times3=83 - 4.5=78.5\)
Upper fence \(=Q_3 + 1.5\times IQR=86+1.5\times3=86 + 4.5=90.5\)

Step2: Check if 76 is an outlier

Since \(76<78.5\)

Answer:

The percentile is 15.

Part b)