According to a research institution, men spent an average of $134.71 on Valentin

Question

According to a research institution, men spent an average of $134.71 on Valentine's Day gifts in 2009. Assume the standard deviation for this population is $40 and that it is normally distributed. A random sample of 10 men who celebrate Valentine's Day was selected. Complete parts a through e.

a. Calculate the standard error of the mean.

b. What is the probability that the sample mean will be less than $125?

c. What is the probability that the sample mean will be more than $145?

d. What is the probability that the sample mean will be between $120 and $160?

e. Identify the symmetrical interval that includes 95% of the sample means if the true population mean is $134.71.

Explanation / Answer

According to a research institution, men spent an average of $134.71 on Valentine's Day gifts in 2009. Assume the standard deviation for this population is $40 and that it is normally distributed. A random sample of 10 men who celebrate Valentine's Day was selected. Complete parts a through e.

standard error = sd/sqrt(n) =40/sqrt(10) =12.6491

Z value for 125, z=(125-134.71)/12.6491 =-0.77

P( mean x <125) = P( z < -0.77) = 0.2206

Z value for 145, z=(145-134.71)/12.6491 =0.81

P( mean x >145) = P( z >0.81) = 0.209

Z value for 120, z=(120-134.71)/12.6491 =-1.16

Z value for 160, z=(160-134.71)/12.6491 =2.0

P( 120<x<160) = P(-1.16<z<2.0)

=P( z <2.0) –P( z < -1.16)

= 0.9772 - 0.123

=0.8542

e. Identify the symmetrical interval that includes 95% of the sample means if the true population mean is $134.71.

z value for symmetrical interval that includes 95% =1.96

interval =( 134.71-1.96*12.6491, 134.71+1.96*12.6491)

=(109.92, 159.502)

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