AP Statistics Topic 4.2: Constructing a Confidence Interval for a Population Mean or Population Mean Difference

01 · Start here

The concept in 60 seconds

A sample mean is an estimate, not the whole answer. If a sample of bottles averages $501.2 \,\mathrm{mL}$ , we can use that value to estimate the population mean fill volume. A confidence interval adds a margin of error to show the uncertainty created by random sampling.

The construction: $\text{estimate}\pm\text{margin of error}$ .

For a population mean with unknown population $\operatorname{SD}$ , use $\bar x\pm t^*\frac{s}{\sqrt n}$ , after checking the conditions. The critical value $t^*$ comes from a $t$ -distribution with $\mathrm{df}=n-1$ .

For paired data, first calculate $1$ difference for each pair. Then construct the same kind of one-sample interval using the mean and $\operatorname{SD}$ of those differences.

This lesson focuses on choosing and constructing the interval. We also practise a clear interpretation; evaluating claims from an interval is developed in Topic 4.3.

02 · One situation, deeper understanding

A bottle-filling investigation

Fictional teaching investigation: A quality-control team wants to estimate the mean fill volume of a lot of $10{,}000$ bottles. It selects an SRS of $25$ bottles without replacement. The sample mean is $501.2 \,\mathrm{mL}$ , and the sample standard deviation is $7.5 \,\mathrm{mL}$ . Previous process information supports an approximately normal population model. The population mean $\mu$ and population $\operatorname{SD}$ $\sigma$ are unknown.

The team wants a $95\%$ confidence interval for $\mu$ , the mean fill volume of all bottles in this lot. It is estimating a mean quantitative measurement, not the proportion of bottles above a cutoff.

Predict: Would $\bar{x} \pm 7.5$ describe the uncertainty of an average properly? Which quantity measures individual variation, and which estimates variation in sample averages?

Visual guide 1: identify the pieces before calculating

Estimate $\bar{x}=501.2 \,\mathrm{mL}$

The sample average estimates the unknown population average.

Estimated sampling spread $\operatorname{SE}=\frac{7.5}{\sqrt{25}}=1.5 \,\mathrm{mL}$

The $\operatorname{SD}$ of individual values is $7.5 \,\mathrm{mL}$ ; the estimated $\operatorname{SD}$ of the sample mean is $1.5 \,\mathrm{mL}$ .

Allow for uncertainty $t^*\approx 2.0639$

At $95\%$ confidence with $\mathrm{df}=24$ , multiply $\operatorname{SE}$ by this critical value.

The confidence level and degrees of freedom determine $t^*$ . The sample statistics determine the center and estimated standard error. Check the sampling design and shape before interpreting the interval.

Topic 4.1 supplied population parameters to study sampling distributions. Here the population parameters are unknown, so we estimate sampling spread with $\frac{s}{\sqrt{n}}$ and use $t$ rather than substituting $1.96$ automatically.

03 · Understand the language

Key ideas and notation

Keep the population target and the sample inputs distinct.
Symbol / term	Meaning	In the bottle investigation
$\mu$	The population mean, a fixed unknown parameter.	Mean fill volume of all $10{,}000$ bottles in this lot.
$\bar{x}$	The sample mean, our point estimate of $\mu$ .	$501.2 \,\mathrm{mL}$ .
$\sigma$ and $s$	Population $\operatorname{SD}$ and sample $\operatorname{SD}$ ; $\sigma$ is unknown here.	Use $s=7.5 \,\mathrm{mL}$ to estimate individual population variability.
$n$ and $N$	Number sampled and number in the population.	$n=25$ bottles; $N=10{,}000$ bottles.
$\operatorname{SE}_{\bar x}$	Estimated standard deviation of the sample mean: $\frac{s}{\sqrt{n}}$ .	$1.5 \,\mathrm{mL}$ .
$t^*$	Positive critical value for the central confidence area of the relevant $t$ -distribution.	About $2.0639$ for $95\%$ and $\mathrm{df}=24$ .
$\mathrm{df}$	Degrees of freedom for a one-sample $t$ procedure.	$n-1=24$ .
$\operatorname{ME}$	Margin of error: $t^* \times \operatorname{SE}$ .	About $3.096 \,\mathrm{mL}$ .
$\mu_d$	Population mean of the paired differences, in a stated order.	For $\text{before}-\text{after}$ completion times, the mean time reduction in the population.
$\bar d$ and $s_d$	Mean and sample $\operatorname{SD}$ of the observed paired differences.	Compute them from the difference list, not from either original list alone.

Why $n-1$ ? Once the sample mean is calculated, the $n$ deviations from that mean must sum to $0$ . Only $n-1$ of those deviations can vary freely. That is the degrees of freedom used in the one-sample $t$ model.

04 · Choose the reference model

Understand $t$ and $t^*$

A $t$ -distribution is symmetric and bell-shaped around $0$ . Compared with the standard normal curve, it has heavier tails, especially at small degrees of freedom. It allows for the extra uncertainty caused by estimating $\sigma$ with $s$ .

The $t$ family contains different curves for different $\mathrm{df}$ values. As $\mathrm{df}$ increases, the curves approach the standard normal curve. Do not claim that every $t$ curve has $\operatorname{SD}$ $1$ . The standardized $t$ model is not an ordinary normal distribution.

For a two-sided $95\%$ interval, the central area is $0.95$ and each tail has area $0.025$ . The positive cutoff $t^*$ therefore has cumulative area $0.975$ to its left.

Visual guide 2: connect heavier tails with a confidence cutoff

      EXCELLENT  
   Based on 51 reviews 
    
 
 
 
   
 
 
       Posted on Google  
   
   0000003998 : Abdad Alam Shamim Alam 
 
 
 
 Trustindex verifies that the original source of the review is Google. This is the bestest institution I have ever come to and I love it very much.
  
 
      Posted on Google  
   
   Aliki S 
 
 
 
 Trustindex verifies that the original source of the review is Google. 
  
 
      Posted on Google  
   
   Ali 
 
 
 
 Trustindex verifies that the original source of the review is Google. Sly is him
  
 
      Posted on Google  
   
   Jitendra Kumar Kumawat 
 
 
 
 Trustindex verifies that the original source of the review is Google. 
  
 
      Posted on Google  
   
   Raahil Hasan 
 
 
 
 Trustindex verifies that the original source of the review is Google. 
  
 
      Posted on Google  
   
   Tina Mudarres 
 
 
 
 Trustindex verifies that the original source of the review is Google. The classes are amazing and my child learnt so much
  
 
      Posted on Google  
   
   alisha gadoya 
 
 
 
 Trustindex verifies that the original source of the review is Google. I have experienced a lot of good things, it has taught me so many things and from B/Cs I have gone to an A! This is wonderful and Mavish’s class is awesome.
  
 
      Posted on Google  
   
   smasher 123 
 
 
 
 Trustindex verifies that the original source of the review is Google. It’s sensational my kid went. First he only would get C now it’s all A’s really good and recommend
  
 
      Posted on Google  
   
   Mosa Al- Samaraie 
 
 
 
 Trustindex verifies that the original source of the review is Google. AMAZING PRICES GREAT TEACHERS STRAIGHT FORWARD LEARNING STEADY PACE IN TUTORING
  
 
      Posted on Google  
   
   Cael Dagnelie 
 
 
 
 Trustindex verifies that the original source of the review is Google. I had a great experience learning math with Ms. Mavish. She explains complex topics in a very clear and simple way, which made it easier for me to understand and enjoy the subject. Her patience and dedication really stood out, and she always made sure that everyone in the class was keeping up. I especially appreciated how approachable she was — I never felt afraid to ask questions, and she was always willing to help. Thanks to her teaching, my confidence in math has grown a lot. I’m really thankful for the effort she puts into every lesson!

NUM8ERS is one of finest tutoring institutes in UAE, Located in Al Barsha 1, Dubai. Close to DUBAI AMERICAN ACADEMY (DAA) & AMERICAN SCHOOL OF DUBAI (ASD).

Constructing a Confidence Interval for a Population Mean or Population Mean Difference

By the end of this lesson, you should be able to:

The concept in 60 seconds

A bottle-filling investigation

Key ideas and notation

Understand $t$ and $t^*$

Main Menu

Quick Links

The concept in 60 seconds

A bottle-filling investigation

Key ideas and notation

Understand tt and t∗t^*

Main Menu

Quick Links​

Understand $t$ and $t^*$

Quick Links