Suppose you are given a population of size 600 with a mean...

Suppose you are given a population of size 600 with a mean of 75 and a standard deviation of 16. If you take a simple random sample of size 35, what are the following values? (Enter z-values to 2 decimals and probabilities to 4 decimals) (Hint: Don't forget to check the value of $\frac{n}{N}$) (a) Calculate the standard error (to 4 decimals). $\sigma_{\bar{x}} =$ (b) Calculate the probability the sample mean will be smaller than 72. (Base the probability on the rounded z-value.) $P(\bar{x} <$ ______ )$= P(z <$ ______ )$=$ ______ (c) Calculate the probability that the sample mean will be at least 77. (Base the probability on the rounded z-value.) $P(\bar{x} \ge$ ______ )$= P(z \ge$ ______ )$=$ ______ You may need to use the appropriate appendix table or technology to answer this question.

Transcript

0:00 Hello everyone.

00:02 We are going to solve a question and in this question we are given that mean is equals to dollar 60 standard deviation is dollar 12 and number of samples and is equals to nine.

00:14 So in a part we have to tell with the small sample size what condition is necessary to apply the central limit theorem.

00:22 So in this we have to tell the necessary condition.

00:25 As the population is, population is normally distributed sample are drawn randomly and independent to each other.

01:00 This is our necessary condition.

01:03 And now moving on to the next part, b part.

01:07 In this, what is the standard error? we have to calculate standard error.

01:12 Standard arrow is equals to the standard deviation divided by under root n, which is equals to 12 divided by under root 9.

01:25 So this is equals to 4.

01:27 So answer is standard error is equal to 4.

01:32 This is our final answer for b part.

01:35 Now moving on to part c, in this we have to calculate the part.

01:40 Probability that the sample mean is greater than dollar 63 that probability mean is greater than 63 so this is equals to p this mean minus mu divided by standard error and this is greater than 63 minus 60 divided by 4 so this p is z greater than 0 .75 so this p is z greater than 0 .75 so this this is equals to 1 minus p z less than 0 .75.

02:16 And using the z score table, this is 1 minus 0 .7734 which is equals to 0 .2266.

02:28 Hence the probability that the mean is greater than 63 is equals to 0 .2266.

02:37 This is the answer of c part.

02:39 So now moving on to the part d.

02:44 In d part we have to calculate the probability that the mean is less than $1 .56 probability that this mean, sorry, here it is, probability that this mean is less than $56.

03:10 So this is equals to p x bar mean minus mu divided by standard error is less than 56 minus 60 divided by 4...

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