Suppose the lengths of the pregnancies of a certain animal...

Suppose the lengths of the pregnancies of a certain animal are approximately normally distributed with mean μ = 297 days and standard deviation σ = 15 days. Complete parts (a) through (f) below. (b) Suppose a random sample of 17 pregnancies is obtained. Describe the sampling distribution of the sample mean length of pregnancies. (c) What is the probability that a random sample of 17 pregnancies has a mean gestation period of 291 days or less? Interpret this probability. Select the correct choice below and fill in the answer box within your choice. (d) What is the probability that a random sample of 41 pregnancies has a mean gestation period of 291 days or less? What might you conclude if a random sample of 41 pregnancies resulted in a mean gestation period of 291 days or less? This result would be expected or unusual, so the sample likely came from a population whose mean gestation period is equal to / less than / greater than 297 days. (f) What is the probability a random sample of size 19 will have a mean gestation period within 9 days of the mean? The probability that a random sample of size 19 will have a mean gestation period within 9 days of the mean is ______

Transcript

00:01 So we have individual gestation periods.

00:04 This is again for individuals.

00:07 Have a mean of 297 days and a standard deviation of 15 days.

00:15 And in part a, you didn't actually give a part a, but part b says if we have a sample size of 17 is selecting, i want to describe what the sampling distribution will look like.

00:28 Well, the sampling distribution will be, since this was approximately normal, this will be approximately normal.

00:35 And it will still center at 297 days.

00:43 However, the standard deviation will end up being 15 divided by the square root of 17.

00:50 And 15, let's see, it's roughly 15 divided by 4, 15 divided by the square root of 17 comes out to be 3 .638 days.

01:03 And it'll be approximately normal.

01:05 Now, part c, we want to find, with that sample size of 17, what's the likelihood that the gestation period of that, though 17 has a mean that is less than or equal to 291? and so we'll convert that to a z value and take that 291 minus the 297 and then divided by at 15 over the square root of 17.

01:31 And let's see what we get.

01:33 That's a negative 6 in the top.

01:36 So negative 6 in the top divided by, and we have the 15 divided by the square root of 17.

01:47 And that z value comes out to be negative, approximately negative 1 .65.

01:59 And the area below, negative 1 .6 .5.

02:03 0 .0 .0495.

02:12 Now on part d, if the sample size were 41, then our sampling distribution would change, and we still want to find what's the likelihood of having it be less than or equal to 291.

02:25 And the only calculation change is this 41, well, this will become 41, and this is going to make our z value be larger.

02:35 And so i'm just going to go back up in second entry and change that to 41 and that z value becomes negative 2 .56 and the area below negative 2 .56 is 0 .052.

02:53 So i think you had a labeling of a question being e.

02:59 You really didn't have a label in the typed question, but i think we would, if you did get this with a sample size of 41, you would question whether the mean was actually 297 or whether this was a random sample, but you would question this because this is very unusual, very rare to happen in a distribution that has the mean be 297.

03:31 It would be very rare to get a sample mean that was less than are equal to 291 from that sample size.

03:39 Then finally on question f, we're back to us, well, we're to a sample size now of 19.

03:47 I'm going to double check on the question and make sure it was 19...

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