Researchers measured the data speeds for a particular smartphone carrier at 50 airports. The highest speed measured was

71.771.7

Mbps. The complete list of 50 data speeds has a mean of

x overbarxequals=16.6216.62

Mbps and a standard deviation of

sequals=17.7317.73

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?

To solve this problem, let's break down each part step by step:

### Given Information:
- Highest data speed: \( 71.7 \, \text{Mbps} \)
- Mean data speed (\( \bar{x} \)): \( 16.62 \, \text{Mbps} \)
- Standard deviation (\( s \)): \( 17.73 \, \text{Mbps} \)

### **Part a: What is the difference between the carrier's highest data speed and the mean of all 50 data speeds?**

We can find the difference by subtracting the mean data speed from the highest data speed.

\[
\text{Difference} = 71.7 - 16.62
\]

Let me calculate this.

\[
\text{Difference} = 55.08 \, \text{Mbps}
\]

So, the difference is \( 55.08 \, \text{Mbps} \).

---

### **Part b: How many standard deviations is the difference found in part (a)?**

To find how many standard deviations the difference represents, we can divide the difference by the standard deviation.

\[
\text{Number of standard deviations} = \frac{\text{Difference}}{s} = \frac{55.08}{17.73}
\]

Let me calculate this.

\[
\text{Number of standard deviations} \approx 3.11
\]

So, the difference is approximately 3.11 standard deviations.

---

### **Part c: Convert the carrier's highest data speed to a z-score.**

The formula for calculating a z-score is:

\[
z = \frac{x - \bar{x}}{s}
\]

Where:
- \( x \) is the data point (highest data speed, \( 71.7 \, \text{Mbps} \))
- \( \bar{x} \) is the mean (\( 16.62 \, \text{Mbps} \))
- \( s \) is the standard deviation (\( 17.73 \, \text{Mbps} \))

Substitute the values:

\[
z = \frac{71.7 - 16.62}{17.73}
\]

Let me calculate this.

\[
z \approx 3.11
\]

So, the z-score is approximately 3.11.

---

### **Part d: Is the carrier's highest data speed significant?**

The z-score of \( 3.11 \) is greater than 2. According to the question, data speeds with z-scores between \(-2\) and \( 2 \) are considered neither significantly low nor significantly high.

Since the z-score of \( 3.11 \) is outside this range, the carrier's highest data speed is **significantly high**.

---

### **Summary of Results:**
- **Part a:** The difference between the highest speed and the mean is \( 55.08 \, \text{Mbps} \).
- **Part b:** The difference is approximately \( 3.11 \) standard deviations.
- **Part c:** The z-score is approximately \( 3.11 \).
- **Part d:** The highest data speed is **significantly high**.

Would you like further details or clarifications on any part? 

Here are 5 related questions to explore:
1. How would the z-score change if the standard deviation were larger?
2. What is the significance of a z-score in statistics?
3. How do z-scores help in comparing different datasets?
4. What would be considered a significantly low data speed?
5. How can we interpret data that lies close to the mean?

**Tip:** Z-scores are an excellent way to compare values from different distributions, as they standardize data by converting it to the same scale.

Researchers measured the data speeds for a particular smartphone carrier at 50 airports. The highest speed measured was 71.7 Mbps. The complete list of 50 data speeds has a mean of 16.62 Mbps and a standard deviation of 17.73 Mbps. a. What is the difference between the carrier's highest data speed and the mean of all 50 data speeds? b. How many standard deviations is that 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 -2 and 2 to be neither significantly low nor significantly high, is the carrier's highest data speed significant?

Explore how to calculate the difference between a smartphone carrier's highest data speed of 71.7 Mbps and the mean speed of 16.62 Mbps. Learn how to compute z-scores, standard deviations, and determine whether this speed is significantly high using key statistical concepts. Perfect for high school statistics learners.

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