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05.08.2020 •
Mathematics
The mean amount purchased by a typical customer at Churchill's Grocery Store is $23.50 with a standard deviation of $5.00. Assume the distribution of amounts purchased follows the normal distribution. For a sample of 50 customers, answer the following questions.
(a) What is the likelihood the sample mean is at least $25.00? (Round z value to 2 decimal places and final answer to 4 decimal places.)
Probability
(b) What is the likelihood the sample mean is greater than $22.50 but less than $25.00? (Round z value to 2 decimal places and final answer to 4 decimal places.)
Probability
(c) Within what limits will 90 percent of the sample means occur? (Round your answers to 2 decimal places.)
Solved
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Ответ:
a.
( to four decimal places)
b.
( to four decimal places )
c. The limits will be between the interval of ( 22.33,24.67 )
Step-by-step explanation:
Given that :
mean = 23.50
standard deviation = 5.00
sample size = 50
The objective is to calculate the following:
(a) What is the likelihood the sample mean is at least $25.00?
Let X be the random variable, the probability that the sample mean is at least 25.00 is:
From the normal tables :
(b) What is the likelihood the sample mean is greater than $22.50 but less than $25.00?
(c) Within what limits will 90 percent of the sample means occur?
At 90 % confidence interval, level of significance = 1 - 0.90 = 0.10
The critical value for the
= 1.65
Standard Error =![\dfrac{\sigma}{\sqrt{n}}](/tpl/images/0717/7796/08572.png)
Standard Error =![\dfrac{5}{\sqrt{50}}](/tpl/images/0717/7796/02963.png)
Standard Error = 0.7071
Therefore, at 90 percent of the sample means, the limits will be between the intervals of :![(\mu \pm z_{\alpha/2} \times S.E)](/tpl/images/0717/7796/bcd17.png)
Lower limit = ( 23.5 - (1.65×0.707) )
Lower limit = ( 23.5 - 1.16655 )
Lower limit = 22.33345
Lower limit = 22.33 (to two decimal places).
Upper Limit = ( 23.5 + (1.65*0.707) )
Upper Limit = ( 23.5 + 1.16655 )
Upper Limit = 24.66655
Upper Limit = 24.67
The limits will be between the interval of ( 22.33,24.67 )
Ответ: