The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 46 o
1 answer:
Answer:
(a) 68% of the widget weights lie between <u>43 ounces</u> and <u>49 ounces</u>.
(b) The percentage of the widget weights lie between 43 and 87 ounces is 15.87%.
(c) The percentage of the widget weights lie below 76 is 100%.
Step-by-step explanation:
Let <em>X</em> = weight of widgets manufactured by Acme Company.
The distribution of the random variable <em>X</em> is, N (<em>μ </em>= 46, <em>σ</em>²<em> </em>=<em> </em>3²).
According to the Empirical Rule in a normal distribution with mean <em>µ</em> and standard deviation <em>σ</em>, nearly all the data will fall within 3 standard deviations of the mean. The empirical rule can be broken into three parts:
- 68% data falls within 1 standard deviation of the mean. That is P (µ - σ ≤ X ≤ µ + σ) = 0.68.
- 95% data falls within 2 standard deviations of the mean. That is P (µ - 2σ ≤ X ≤ µ + 2σ) = 0.95.
- 99.7% data falls within 3 standard deviations of the mean. That is P (µ - 3σ ≤ X ≤ µ + 3σ) = 0.997.
(a)
According to the Empirical rule, 68% data falls within 1 standard deviation of the mean.
P (µ - σ ≤ X ≤ µ + σ) = 0.68.
Compute the upper and lower values as follows:
<em>µ</em> - <em>σ</em> = 46 - 3 = 43 ounces
<em>µ</em> + <em>σ</em> = 46 + 3 = 49 ounces
Thus, 68% of the widget weights lie between <u>43 ounces</u> and <u>49 ounces</u>.
(b)
Compute the probability of the widget weights lie between 43 and 87 ounces as follows:
*Use a <em>z</em>-table.
The percentage is, 0.1587 × 100 = 15.87%.
Thus, the percentage of the widget weights lie between 43 and 87 ounces is 15.87%.
(c)
Compute the probability of the widget weights lie below 76 as follows:
*Use a <em>z</em>-table.
The percentage is, 1 × 100 = 100%.
Thus, the percentage of the widget weights lie below 76 is 100%.
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Answer:
We accept the null hypothesis and reject the alternate hypothesis. There is no evidence to conclude that the population mean is greater than 29. The population mean is less than or equal to 29.
Step-by-step explanation:
We are given the following in the question:
Population mean, μ = 29
Sample mean, = 30
Sample size, n = 47
Alpha, α = 0.05
Population standard deviation, σ = 5
First, we design the null and the alternate hypothesis
a) This is a one-tailed test because the alternate hypothesis is in greater than direction.
We use One-tailed z test to perform this hypothesis.
b)
, we reject the null hypothesis and accept the alternate hypothesis and if
, we accept the null hypothesis and reject the alternate hypothesis.
c) Formula:
Putting all the values, we have
d) Now,
Since,
We accept the null hypothesis and reject the alternate hypothesis. There is no evidence to conclude that the population mean is greater than 29. The population mean is less than or equal to 29.
e) P-value is 0.0853
On the basis of p value we again accept the null hypothesis.