When to use t-distribution when calculating a confidence interval

probabilitystatistics

I have two exercises:

  1. The duration of the charge of a battery from a certain laptop brand follows a normal distribution with stdev of half an hour. In a random
    sample of 15 laptops, the average duration of the battery was 5.8
    hours. Build a confidence interval of 95% for the average duration of
    the battery.

  2. The research department of a tire manufacture is investigating the duration of a tire using a new rubber component. 16 tires were
    produced and the duration of each was tested. The average duration and
    the stdev were 60139.7 and 3645.94 Km, respectively. Admitting that
    the average duration of the tire follows a normal distribution,
    determine a 95% confidence interval for the average.

My professor solved them like this:

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("Tabela" refers to z tables)

I understand why he solved the first one that way, I don't understand why he did that on the second one. I solved it like the first one and got similar results. From the book we are using:

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When do I use one and when do I use the other? Both the exercises and the formulas look identical to me.

Best Answer

The difference is a question of whether the standard deviation is estimated from the sample, or if the true value is known and fixed regardless of the data. When it is known and fixed, we use the critical value for a standard normal distribution. When the true standard deviation is not known and is estimated from the data, then we use the critical value for a Student's $t$ distribution.

As you can see from the first question, you are told the standard deviation is half an hour. This is not calculated from the sample; it is given to you in advance. In the second question, you are given an estimate of the standard deviation from the sample of 16 tires. Even though the data are in each case are normally distributed, the sampling distribution of the sample mean is not the same: in the first case, it is normal; but in the second, it is Student's $t$ with $n-1 = 15$ degrees of freedom.

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