Question

It is known that the weight of a typical cellphone is normally distributed with a known standard deviation. In a random sample of 15 cell phones, the sample mean weight was 155.7 g. Moreover, it is known that (146.59 ,164.81) is a 95% confidence interval for the true mean weight of this type of cell phones.
a) Find the value of the point estimate of the mean weight of all cell phones of this type.
b) Construct a 90% confidence interval for the true mean weight of this type of cell phones.
c) Interpret the resulting confidence interval in part (b).
d) Suppose the confidence interval obtained in part (b) is too wide. How can the width of this interval be reduced? Discuss all possible alternatives. Which alternative is the best?
e) How many cell phones must we select so that the length of a 94% confidence interval is 20.
f) A random sample of 25 cell phones from this population is picked which yields a mean of 150 g. If the maximum probability of a Type I error is 0.02, can we conclude that the true mean weight of this type of cell phones is less than $155 g? Use the critical value approach.
g) At a 1% significance level and using the ?-value approach, test the hypothesis in part (f).

Answer 1

a)

Point estimate of the mean weight of all cell phones of this type = Sample mean = 155.7 g.

b)

Margin of error E = (164.81 - 146.59)/2 = 9.11

z value for 95% confidence interval is 1.96

Standard error of mean = E / z = 9.11 / 1.96 = 4.647959

z value for 90% confidence interval is 1.645

90% confidence interval for the true mean weight of this type of cell phones is,

(155.7 - 1.645 * 4.647959, 155.7 + 1.645 * 4.647959)

(148.0541 ,  163.3459)

c)

There is a 90% chance that the confidence interval (140.4082 ,  170.9918) contains the true mean weight of this type of cell phones.

d)

We can reduce the width by decreasing the confidence level or increasing the sample size.

Increasing the sample size is best as we can construct a narrower confidence interval for any confidence level.

e)

Margin of error E = 20/2 = 10

z value for 94% confidence interval is  1.88

= * Standard error of mean = * 4.647959 = 18.00147

Sample size, n = (z   / E)²

= (1.88 * 18.00147 / 10)²

= 11.45   12

f)

Standard error of mean = / = 18.00147 / =  3.600294

Test statistic, z = (150 - 155) / 3.600294 = -1.39

Critical value of z at 0.02 significance level is -2.05

Since the test statistic is not less than -2.05,  there is no strong evidence that true mean weight of this type of cell phones is less than $155 g.

g)

P-value = P(z < -1.39) = 0.0823

Since, p-value is greater than 0.01 significance level, we conclude that there is no strong evidence that true mean weight of this type of cell phones is less than $155 g.