QuestionAccording to an online source, the mean time spent on smartphones daily by adults in a country is 2.55 hours. Assume that this is correct and assume the standard deviation is 1.2 hours. Complete parts (a) and (b) below.
a. Suppose 150 adults in the country are randomly surveyed and asked how long they spend on their smartphones daily. The mean of the sample is recorded. Then we repeat this process, taking 1000 surveys of 150 adults in the country. What will be the shape of the distribution of these sample means?
The distribution will be because the values will be
Studdy Solution
STEP 1
What is this asking? If we repeatedly survey 150 people about their phone time and calculate the average for each survey, what will the distribution of these averages look like? Watch out! Don't confuse the distribution of *individual* phone times with the distribution of *sample means*.
STEP 2
1. Check if the Central Limit Theorem applies.
STEP 3
We're taking **150** samples each time.
Since , our sample size is large enough for the Central Limit Theorem to apply!
This theorem is like a magical key that unlocks the shape of the distribution of sample means.
STEP 4
The Central Limit Theorem says that if we take many large enough samples from *any* distribution, the distribution of the sample means will be approximately **normal**.
It's a superpower!
It doesn't matter what the original distribution of individual phone times looks like (we don't even know!), the distribution of the *averages* of those phone times will be approximately normal.
STEP 5
Because our sample size is large enough, the Central Limit Theorem tells us that the distribution of these sample means will be approximately **normal**!
STEP 6
The distribution will be **normal** because the Central Limit Theorem applies since .
The values will be the **sample means** of smartphone usage from each survey.
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