Math  /  Data & Statistics

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From a sample of 16 students in our class, the mean resting heart rate was 68 bpm with a standard deviation of 12 bpm . Assuming a normal distribution, use the erripirical rule to calculate the following. What is the 95%95 \% confidence interval for the average heart rate of all students in our class? (66,70)(66,70) (60,72)(60,72) (62,74)(62,74) (58,76)(58,76) Question 2 1 pts Accessibility Resources
For a group of normally distributed scores with a mean of 200 and a standard deviation of 10,

Studdy Solution

STEP 1

What is this asking? We need to find the range of heart rates where we're 95% sure the *true* average heart rate of the *whole* class falls, based on a small sample of 16 students. Watch out! Don't mix up the standard deviation of the *sample* with the standard error of the *mean*.
We need the standard error to build the confidence interval!

STEP 2

1. Calculate the standard error.
2. Calculate the margin of error.
3. Calculate the confidence interval.

STEP 3

The **standard error** tells us how much the average of our sample is likely to be off from the *true* average of the whole class.
It's like a measure of how wobbly our sample average is.
We find it by dividing the **standard deviation** of the sample, which is σ=12\sigma = 12, by the square root of the **sample size**, which is n=16n = 16.

STEP 4

So, the **standard error** is: SE=σn=1216=124=3 SE = \frac{\sigma}{\sqrt{n}} = \frac{12}{\sqrt{16}} = \frac{12}{4} = 3 This means our sample's average heart rate could be off by about 3 bpm, compared to the *true* class average.

STEP 5

The **margin of error** tells us how far we need to go above and below our sample average to be 95% confident we've captured the *true* average.
For a 95% confidence interval, we multiply the **standard error** by **1.96** (this comes from the normal distribution – about 95% of the data falls within 1.96 standard deviations of the mean).

STEP 6

Our **margin of error** is: ME=1.96SE=1.963=5.88 ME = 1.96 \cdot SE = 1.96 \cdot 3 = 5.88 So, we need to go about 5.88 bpm above and below our sample average.

STEP 7

The **confidence interval** is the range where we're pretty sure the *true* average lies.
We find it by taking our **sample mean** and adding/subtracting the **margin of error**.
Our **sample mean** is xˉ=68\bar{x} = 68.

STEP 8

The **lower bound** of our interval is: xˉME=685.88=62.12 \bar{x} - ME = 68 - 5.88 = 62.12 The **upper bound** is: xˉ+ME=68+5.88=73.88 \bar{x} + ME = 68 + 5.88 = 73.88

STEP 9

So, our 95% confidence interval is approximately (62.12,73.88)(62.12, 73.88).
Looking at the available options, (62,74)(62, 74) is the closest match!

STEP 10

The 95% confidence interval for the average heart rate is approximately (62,74)(62, 74).

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