A manufacturer of cereal would like to know whether its bag filling machine works correctly at the 600 gram setting. It is believed that the machine is overfilling the bags. A 9 bag sample had a mean of 615 grams with a variance of 169. Assume the population is normally distributed. Is there sufficient evidence at the 0.01 level that the bags are overfilled?

Answer:
Given:

The Hypothesized Mean (μ)=600
The Sample Mean (x¯)=615
The Sample Variance (s2)=169
The Sample Size (n)=9

The Sample Standard Deviation (s)=169=13
The Significance Level (α)=0.01

Solution:
The null and alternative hypothesis:
H0:μ=600
H1:μ>600

The test statistic (t):
t=x¯μsn
=615600139

=3.462

The degree of freedom (df):
df=n1
=91
=8

The p-value:
p-value=P(t8>3.462)

=0.0043

The conclusion:
The p-value is less than the significance level. Therefore, we reject the null hypothesis. There is sufficient evidence to support the claim that the bags are overfilled.

Final Answer:
The null and alternative hypothesis:
H0:μ=600
H1:μ>600

The test statistic (t)=3.462

The p-value =0.0043

The conclusion:
The p-value is less than the significance level. Therefore, we reject the null hypothesis. There is sufficient evidence to support the claim that the bags are overfilled.

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