Question

The consumer magazine also claims that the cinnamon rolls at STARBUCKS do not weigh at least 8 ounces. A random sample of 25 customers purchasing cinnamon rolls yields the following results: - the sample mean equals 7.87 ounces - it is known from previous studies that the population standard deviation equals 0.25 ounces.

a. Set up a 95% confidence interval for the true mean?

b. What sample size is required if you want to be 99% sure that the sample mean will be within 0.2 ounces of the true mean?

c. Test the hypothesis that the true population mean is less than 8 ounces. Set the type one error equal to 1%.

Answer 1

a)

sample mean, xbar = 7.87
sample standard deviation, σ = 0.25
sample size, n = 25

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

ME = zc * σ/sqrt(n)
ME = 1.96 * 0.25/sqrt(25)
ME = 0.098

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (7.87 - 1.96 * 0.25/sqrt(25) , 7.87 + 1.96 * 0.25/sqrt(25))
CI = (7.77 , 7.97)

b)

The following information is provided,
Significance Level, α = 0.01, Margin or Error, E = 0.2, σ = 0.25

The critical value for significance level, α = 0.01 is 2.58.

The following formula is used to compute the minimum sample size required to estimate the population mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (2.58 * 0.25/0.2)^2
n = 10.4

Therefore, the sample size needed to satisfy the condition n >= 10.4 and it must be an integer number, we conclude that the minimum required sample size is n = 11
Ans : Sample size, n = 11 or 10

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if we take z value upto 3 or 4 decimal answer would be chnage

c)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 8
Alternative Hypothesis, Ha: μ < 8

Rejection Region
This is left tailed test, for α = 0.01
Critical value of z is -2.326.
Hence reject H0 if z < -2.326

Test statistic,
z = (xbar - mu)/(sigma/sqrt(n))
z = (7.87 - 8)/(0.25/sqrt(25))
z = -2.6

P-value Approach
P-value = 0.0047
As P-value < 0.01, reject the null hypothesis.