Question

talk2u and proze are two social media apps in which users can post messages in text. the character counts for posts on talk2u have a population mean of 68 and a standard deviation of 26. the character counts for posts on proze have a population mean of 453 and a standard deviation of 21. for each app, the distribution of the character counts is clearly bell - shaped. manuels last post on talk2u was 151 characters. alans last post on proze was 503 characters. (a) find the z - scores of manuels last post as a post on talk2u and alans last post as a post on proze. round your answers to two decimal places. z - score of manuels post: \square z - score of alans post: \square (b) relative to its population, which post was longer? choose the best answer based on the z - scores of the two posts. \bigcirc manuels post \bigcirc alans post \bigcirc it is unclear which post was longer relative to its population

Explanation:

Part (a)

The formula for the z - score is $z=\frac{x - \mu}{\sigma}$, where $x$ is the value from the dataset, $\mu$ is the population mean, and $\sigma$ is the population standard deviation.

Step 1: Calculate the z - score for Manuel's post

For Manuel's post on Talk2U:

• $x = 151$ (the number of characters in Manuel's post)
• $\mu=68$ (the population mean for Talk2U)
• $\sigma = 26$ (the population standard deviation for Talk2U)

Using the z - score formula:
$z=\frac{151 - 68}{26}=\frac{83}{26}\approx3.19$

Step 2: Calculate the z - score for Alan's post

For Alan's post on Proze:

• $x = 503$ (the number of characters in Alan's post)
• $\mu = 453$ (the population mean for Proze)
• $\sigma=21$ (the population standard deviation for Proze)

Using the z - score formula:
$z=\frac{503 - 453}{21}=\frac{50}{21}\approx2.38$

Part (b)

To determine which post is longer relative to its population, we compare the z - scores. A higher z - score means that the value is further above the mean (in terms of standard deviations) in its respective distribution.

Manuel's z - score is approximately $3.19$ and Alan's z - score is approximately $2.38$. Since $3.19>2.38$, Manuel's post is further above the mean of its population than Alan's post is above the mean of its population. So Manuel's post is longer relative to its population.

We compare the z - scores. Manuel's z - score (3.19) is greater than Alan's z - score (2.38). A higher z - score indicates that the post is further above the mean of its population.

Answer:

s:
(a)

• z - score of Manuel's post: $\boldsymbol{3.19}$
• z - score of Alan's post: $\boldsymbol{2.38}$

(b)