Researchers measured the data speeds for a particular smartphone carrier at 50 airports. The highest speed measured was 71.4
 
Mbps. The complete list of 50 data speeds has a mean of x overbar
equals15.39 
Mbps and a standard deviation of sequals
32.15
 
Mbps.
a. What is the difference between​ carrier's highest data speed and the mean of all 50 data​ speeds?
b. How many standard deviations is that​ [the difference found in part​ (a)]?
c. Convert the​ carrier's highest data speed to a z score.
d. If we consider data speeds that convert to z scores between minus
2 
and 2 to be neither significantly low nor significantly​ high, is the​ carrier's highest data speed​ significant?

Let's go through each part of the problem one by one:

### Given information:
- Highest data speed: \( 71.4 \, \text{Mbps} \)
- Mean of data speeds: \( \bar{x} = 15.39 \, \text{Mbps} \)
- Standard deviation: \( s = 32.15 \, \text{Mbps} \)

### a. What is the difference between the​ carrier's highest data speed and the mean of all 50 data​ speeds?
The difference between the highest data speed and the mean is:
\[
\text{Difference} = 71.4 - 15.39 = 56.01 \, \text{Mbps}
\]

### b. How many standard deviations is that​ [the difference found in part​ (a)]?
To find how many standard deviations the difference represents, we divide the difference by the standard deviation:
\[
\frac{\text{Difference}}{\text{Standard Deviation}} = \frac{56.01}{32.15} \approx 1.74
\]
Thus, the difference is approximately \( 1.74 \) standard deviations from the mean.

### c. Convert the​ carrier's highest data speed to a z score.
The z-score is calculated as:
\[
z = \frac{\text{Value} - \text{Mean}}{\text{Standard Deviation}} = \frac{71.4 - 15.39}{32.15} = \frac{56.01}{32.15} \approx 1.74
\]
So, the z-score is \( 1.74 \).

### d. If we consider data speeds that convert to z scores between \(-2\) and \(2\) to be neither significantly low nor significantly​ high, is the​ carrier's highest data speed​ significant?
Since the z-score is \( 1.74 \), which falls between \(-2\) and \(2\), the carrier's highest data speed is not considered significant. It is within the range of typical variation.

### Recap:
a. The difference between the highest speed and the mean is \( 56.01 \, \text{Mbps} \).  
b. The difference is approximately \( 1.74 \) standard deviations.  
c. The z-score for the highest data speed is \( 1.74 \).  
d. The highest data speed is not statistically significant since the z-score is within the typical range.

Would you like more details or have any questions?

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Here are 5 related questions for further exploration:
1. What would the z-score be if the highest data speed were 90 Mbps?
2. How would the result change if the standard deviation were smaller, say 20 Mbps?
3. What does it mean for a data point to have a negative z-score?
4. How would the interpretation change if the z-score threshold for significance were tighter, say between -1.5 and 1.5?
5. How could outliers affect the calculation of the mean and standard deviation?

**Tip:** Z-scores are useful for comparing values from different datasets by standardizing the data relative to the mean and standard deviation.

Researchers measured the data speeds for a particular smartphone carrier at 50 airports. The highest speed measured was 71.4 Mbps. The complete list of 50 data speeds has a mean of 15.39 Mbps and a standard deviation of 32.15 Mbps. a. What is the difference between​ carrier's highest data speed and the mean of all 50 data​ speeds? b. How many standard deviations is that​ [the difference found in part​ (a)]? c. Convert the​ carrier's highest data speed to a z score. d. If we consider data speeds that convert to z scores between minus 2 and 2 to be neither significantly low nor significantly​ high, is the​ carrier's highest data speed​ significant?

Explore how to calculate the z-score and assess the significance of a smartphone carrier's data speed at airports. Learn to apply statistical concepts such as mean, standard deviation, and z-scores. This example uses real data with a mean of 15.39 Mbps, a standard deviation of 32.15 Mbps, and a maximum speed of 71.4 Mbps.

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