Confidence intervals problems for practice and concept clarity

 Confidence intervals problem 1

Question (Confidence intervals problems):  From studies conducted prior it is known that under specified conditions the time taken for a particular chemical reaction to take place is normally distributed. Alicia has conducted the experiment 42 times and found the 90% confidence interval for average time for chemical reaction as [53 secs, 57 secs].

  1. Build a 99% confidence interval for average chemical reaction time.

 

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 =Solution:

90% confidence interval limits are 53 and 57.

Concept: Confidence interval is given by

cil1-1

 

Here, we will use z as the distribution is given as normal and sample size is high (42)

Sample mean – 1.64 * σ/√42 = 53

Sample mean + 1.64 * σ/√42 = 57

Hence the sample mean is 55 secs and σ is 7.90 secs

99% Confidence interval is given by

Sample mean (55) – 1.96 * σ/√42 = 52.60 secs

and Sample mean (55) + 1.96 * σ/√42 = 57.39 secs


Confidence intervals problem 2

Question (Confidence intervals problems): A popular social media platform claims that on an average each of its registered users logs into their profiles for about 30 times in a year. If the number of login data for a random sample of 144 users is collected and analyzed it is found that the average login count in a year is 26.

Assume that the population standard deviation is 3.1

Calculate the 95% confidence interval for population mean using the sample mean.

Check if the company claimed mean falls in the above interval

 

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 Solution:

Let the parameters of population be µ (mean) and σ (standard deviation)

Here the sample size is large (> 30). Hence, from CLT (Central limit theorem) we can assume that the distribution of sample mean is approximately normal with mean µ and standard deviation σ/SQRT(n)

SQRT ~ Square root n is sample size.

Confidence interval for estimating population mean is given by below formula

Sample mean +/-  Margin of error

Zα/2 at 95% confidence interval is 1.96 (From normal distribution tables)

Margin of error = Zα/2 * σ/SQRT(n) = 1.96 * (3.1)/SQRT(144) = 1.96 * (3.1)/12

MOE = 0.5

Confidence interval lower limit = 26 + 0.5 = 26.5

Upper limit = 26 – 0.5 =25.5

Confidence interval is [25.5, 26.5]

As we can observe the claimed mean of 30 is out of the calculated confidence interval range.


Confidence intervals problem 3

Question (Confidence intervals problems): Assume the time taken by racers to finish the running race (100 mts) follows a normal distribution. If you want to estimate the average time taken by racers to finish the 100 mts race with 90% confidence level and you want to express the interval within +/- 1 secs, what is the minimum sample size that you need to consider? From the past studies the standard deviation for time taken by racers is 2.1 secs.

 

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Solution:

If you wish to revise the concepts, refer to Confidence intervals and Calculating confidence intervals

Confidence interval is given by 

cil1-1

 

Margin of error = Zα/2 * σ/√n = 1 secs

Zα/2  for 90% confidence level is 1.64. 

σ (population standard deviation) = 2.1 secs

1.64 * 2.1/√n  <= 1

√n >= 1.64*2.1/1

n>= 11.86

Hence, the minimum sample size required is 12


 

Confidence intervals problem 4

Question (Confidence intervals problems):

A research firm wants to estimate the average time (Industry average) of a client call. A random sample of 60 calls made by sales teams from different companies is analyzed. Average time for the sample is found to be 3.4 minutes. If the standard deviation of population is known to be 0.4 minutes,

Estimate the 99% confidence interval for the average time of client call and interpret this interval.

 

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Solution:

Sample Size is large enough to assume that the sample mean follows a normal distribution (From Central limit theorm)

Sample mean = 3.4

n = 60

Confidence interval = Sample mean +/-  Margin of error

Zα/2 at 99% confidence interval is 2.58 (From normal distribution tables)

Margin of error = Zα/2 * σ/SQRT(n) = 2.58 * (0.4)/SQRT(60) = 2.58 * (0.4)/7.745

MOE = 0.13

Lower limit = 3.4 – 0.13 = 3.27

Upper limit = 3.4 + 0.13 = 3.53

99% confidence interval for average time of client call is [3.27, 3.53]

Interpretation:

Once the sampling is done and the confidence interval is constructed, confidence levels cannot be seen as probability.

If repeated samples of size 60 are drawn several times and each time confidence interval is constructed around the sample size, 99% of such intervals contain the true population mean.

We are 99% confident that the interval [3.27, 3.53] contains the true average client call time.