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According to the United States Census Bureau, the average single-family home constructed in 2016 was 2,463 square feet. Carmen wondered if families in her area still want a smaller home in comparison to the average. She randomly selected 49 people and asked what the ideal square footage of a home is for them. From the responses, she calculated the sample mean equals 1,640 square feet with a sample standard deviation of 66 square feet. Calculate the value of the test statistic.

Respuesta :

Using the t-distribution, it is found that the value of the test statistic is t = -87.3.

At the null hypothesis, it is tested if the mean is of 2,463 square feet, that is:

[tex]H_0: \mu = 2463[/tex]

At the alternative hypothesis, it is tested if the mean is smaller, that is:

[tex]H_1: \mu < 2463[/tex].

We have the standard deviation for the sample, hence, the t-distribution is used to solve this question.

The test statistic is given by:

[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]

The parameters are:

  • [tex]\overline{x}[/tex] is the sample mean.
  • [tex]\mu[/tex] is the value tested at the null hypothesis.
  • s is the standard deviation of the sample.
  • n is the sample size.

The values of the parameters are: [tex]\overline{x} = 1640, \mu = 2463, s = 66, n = 49[/tex].

Hence, the value of the test statistic is:

[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]

[tex]t = \frac{1640 - 2463}{\frac{66}{\sqrt{49}}}[/tex]

[tex]t = -87.3[/tex]

To learn more about the t-distribution, you can take a look at https://brainly.com/question/25819230

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