# Using 64 randomly selected phone calls, the average call length was calculated t

Using 64 randomly selected phone calls, the average call length was calculated to be 4.2 minutes. It is known from previous studies that the variance of the length of phone calls is $$\displaystyle{1.44}\min^{{{2}}}$$. Assuming that the length of calls has a normal distribution
a) estimate an interval estimate of the length of a telephone conversation at the 0.95 confidence level
b) confidence 0.99
c) Compare the length of the two intervals and explain how the length of the interval depends on the confidence level.

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Step 1
As per given by the question, there are 64 randomly selected phone calls, the average call length was calculated to be 4.2 minutes.
So,
Random value (n) is 64, Mean $$\displaystyle\overline{{{X}}}$$ is 4.42 minute, and Variance is 1.44.
Then first calculate the standard deviation with the help of means and variance.
$$\displaystyle{V}={\frac{{\sigma}}{{\overline{{{X}}}}}}$$
$$\displaystyle{1.44}={\frac{{\sigma}}{{{4.42}}}}$$
$$\displaystyle\sigma={6.36}$$
Step 2
Now,
a) estimate an interval estimate of the length of a telephone conversation at the 0.95 confidence level.
From formula of normal distribution,
$$\displaystyle{C}{I}=\overline{{{X}}}\pm{t}_{{{z}}}\cdot{\frac{{\sigma}}{{\sqrt{{{n}}}}}}$$
$$\displaystyle={4.42}\pm{1.96}\cdot{\frac{{{6.36}}}{{\sqrt{{{64}}}}}}$$
$$\displaystyle={4.42}\pm{\left({1.96}\right)}{\left({0.79}\right)}$$
$$\displaystyle={4.42}\pm{1.44}$$
Hence, the length of a telephone conversation at the 0.95 confidence level is 5.97, 2.87
Step 3
b). estimate an interval estimate of the length of a telephone conversation at the 0.99 confidence level.
From formula of normal distribution,
$$\displaystyle{C}{I}=\overline{{{X}}}\pm{t}_{{{z}}}\cdot{\frac{{\sigma}}{{\sqrt{{{n}}}}}}$$
$$\displaystyle={4.42}\pm{2.58}\cdot{\frac{{{6.36}}}{{\sqrt{{{64}}}}}}$$
$$\displaystyle={4.42}\pm{\left({2.58}\right)}{\left({0.79}\right)}$$
$$\displaystyle={4.42}\pm{2.038}$$
Hence, the length of a telephone conversation at the 0.99 confidence level is 6.458, 2.382.
Step 4
c). Compare the length of the two intervals and explain how the length of the interval depends on the confidence level.
The length of interval at 0.95 confidence level is 5.97, 2.87. and,
The length of interval at 0.99 confidence level is 6.458, 2.382.
Here, the value of length of interval at 0.99 confidence level is 6.458, 2.382, that is greater than the value of length of interval at 0.95 confidence level is 5.97, 2.87.
The length of the interval depends on the confidence level, because if confidence level is increase then length of interval is increase, if confidence level is decrease then length of interval is increase.