Math  /  Data & Statistics

QuestionHow smart is your phone? A random sample of 8 Samsung Galaxy smartphones being sold over the internet in 2013 had the following prices, in dollars: \begin{tabular}{llllll} \hline 149 & 135 & 249 & 349 & 299 & 249 \\ 199 & 169 & & & & \\ \hline \end{tabular} Send data to Excel
Assume the population standard deviation is σ=85\sigma=85. Perform the following.
Part: 0/30 / 3
Part 1 of 3 (a) Explain why it is necessary to check whether the population is approximately normal before constructing a confidence interval.
It is necessary to check whether the population is approximately normal because (Choose one) Skip Part Check Save For Later Submit Ass - 2024 McGraw Hill LLC. All Rights Reserved. Terms of Use I Privacy Center

Studdy Solution

STEP 1

What is this asking? We need to explain why we must check if the smartphone prices are normally distributed before we can calculate a confidence interval. Watch out! Don't confuse sample distributions with population distributions!

STEP 2

1. Confidence Intervals and Normality
2. The Central Limit Theorem (CLT)
3. Small Sample Sizes

STEP 3

Confidence intervals are like giving a best guess for where the *true average* price hides, plus a margin of error.
We're saying, "We're pretty sure the average price of *all* Samsung Galaxy phones is somewhere in this range."

STEP 4

Many statistical methods, like creating confidence intervals, rely on the assumption of normality.
This means the data needs to roughly follow a bell curve shape.

STEP 5

The Central Limit Theorem is a real lifesaver!
It says that even if the prices themselves aren't perfectly normal, the *average* of many samples will start to look like a bell curve.
Imagine taking tons of samples of 8 phones each and calculating their average price.
Those averages would form a normal distribution.

STEP 6

The CLT is especially helpful when we have a large sample size.
The larger the sample, the closer the sample means get to a normal distribution, regardless of the original population's shape.

STEP 7

Here's the catch: with a *small* sample size (like our 8 phones), the CLT isn't as powerful.
If the original population of phone prices is weirdly shaped (like, super skewed or has multiple peaks), then our small sample might not accurately represent the true population.

STEP 8

In our case, if the population of phone prices isn't approximately normal, our confidence interval might be wonky and not very reliable.
It wouldn't give us a good estimate of the *true* average price.

STEP 9

It's necessary to check for normality because with a small sample size like ours (n=8n = 8), the Central Limit Theorem doesn't guarantee a normal sampling distribution unless the population itself is approximately normal.
If the population isn't normal, our confidence interval calculations could be inaccurate.

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