New Automobile Ownership According to a survey, the average...

New Automobile Ownership According to a survey, the average length of time a person keeps a new automobile is 71.7 months. The average time 30 randomly selected automobile owners kept their new automobiles was 70.8 months with ( sigma=4.1 ) months. At the 0.01 level of significance, can it be concluded that for this type of automobile, the owners kept their new automobile less than 71.7 months? Assume that the population is normally distributed. Use the ( P )-value method with tables. Part: 0 / 5 Part 1 of 5 (a) State the hypotheses and identify the claim. [ egin{array}{l} H_{0}: square( ext { Choose one } abla \ H_{1}: square( ext { Choose one } abla end{array} ] The hypothesis test is a ( square ) (Choose one) test.

Transcript

00:01 We would like to perform a hypothesis test.

00:03 We need to start by stating the hypotheses.

00:06 So the null hypothesis always gets some kind of equal sign.

00:09 It represents no change, no difference.

00:12 The alternative doesn't get an equal sign.

00:14 So a survey claims the average length of time a person keeps an automobile is 71 .7 months.

00:21 We want to know if actually it's less than that for this type.

00:26 Okay, less than.

00:27 So specifically, mean is less than.

00:30 The 71 .7 months reference value.

00:34 The null would be it's not less than.

00:37 So you've only got equals tips, we'll put equals.

00:40 Some courses would make you always have the opposite signs.

00:45 So our sample size is 30.

00:48 The sample mean was 70 .8 and the population standard deviation sigma is known to be 4 .1.

00:58 Now we're going to assume the null hypothesis is true.

01:01 Pretend the mean is 71 .7.

01:03 If i took every sample of size 30, took the means, plotted them out, i'd get something approximately normal.

01:11 Thanks to the central limit theorem, as sample size increases, sample means get more and more normally distributed.

01:18 But this one is normally distributed, the population, so you can't get more normal than normal.

01:24 Any sampling distribution based on it, it's also normal.

01:27 Sample means follow normal curve.

01:30 The mean is the same as the population mean.

01:32 Their standard deviation is sigma over root n.

01:36 What does my sample fall on this curve? maybe it's here.

01:40 It's less than 71 .7, but this isn't unlikely.

01:43 It's plausible that the null hypothesis is true, and we just happen to get this difference due to sampling error.

01:49 Or maybe my sample mean is out here.

01:51 And then i would say, if the null hypothesis is true, this is really unlikely.

01:57 Therefore, i don't think it's true.

01:59 Alpha, the level of significance, tells you how unlikely something, has to be before you reject null hypothesis.

02:07 Here, 1%.

02:08 The p -value is the probability of getting our result or something more extreme if the null hypothesis is true.

02:19 So if the p -value ends up being less than or equal to alpha, we're saying, okay, that was 1 % or less likely to happen, rejects the null hypothesis.

02:28 If the p -value is greater than alpha, it wasn't that unlikely.

02:32 This is a left -tailed test.

02:35 I only care if my sample mean is lower than 71 .7...

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