For problems 45 assume the distribution of birth weights of...

For problems 4–5, assume the distribution of birth weights of male infants is normally distributed with a mean of 3100 grams and a standard deviation of 680 grams Suppose a simple random sample of 18 male infants is performed. What would be the mean and standard deviation of this distribution of sample means? (round your final answer for standard deviation to 1 decimal place)

Transcript

00:01 Hello students, in this question we have given the mean of the normal distribution that is mu is equal to 3290 and standard deviation sigma is equal to 520 and now by using this we have to calculate the percentage of the newborns that weigh between 3000 and 4000 grams so that means we have to calculate probability of 3000 is less than x is less than 4000.

00:34 Now here by using the normal approximation we can write this probability as probability of 3000 minus 3290 divided by 520 it is less than x minus mu divided by sigma is less than 4000 minus 3290 divided by 520 and now we know that x minus mu by sigma is nothing but standard normal variable that is z so this is probability of minus 0 .5577 that is less than z is less than 1 .3654 and now this is equal to probability of z is less than 1 .3654 minus probability of z is less than minus 0 .5577 and here by using the z table or standard normal table we will get the probabilities 0 .9147 minus 0 .2877 and that is equal to 0 .6269 so this is the required probability and that is approximately equal to 62 .69 percentage then in the next question we have to find the weight in the grams that would separate the smallest 4 percent of the weights of newborns from the rest so that means we have given probability of x is less than suppose p4 is equal to 0 .04 and we have to calculate the value of p4 so now here by using the normal approximation this is probability of z is less than p4 minus 3290 divided by 520 and that is equal to 0 .04 so now by using the z table we have probability of z less than minus 1 .7507 that is equal to 0 .04 and now if we compare these two expressions then we will get the value of p4 minus 3290 divided by 520 is equal to minus 1 .7507 and that is after rearranging the terms p4 is equal to minus 1 .7507 into 520 plus 3290 and which is equal to 2379 .636 so this is the required value of p4...

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