Question

Suppose a company makes skateboard wheels with a diameter of 64 mm. When the machine that manufactures the wheels is functioning properly, it produces wheels whose diameters are normally distributed with a mean of 64 mm and a standard deviation of 3.5 mm. The machine gets out of tolerance over time, and can produce wheels that have either a larger or smaller mean diameter. Therefore, quality control inspectors randomly select wheels and measure their diameters. A recent random sample of 49 wheels had a mean of 65 mm.

Represent the p-value of the sample results.

Answer 1

Explanation:

We would set up the hypothesis test. This is a test of a single population mean since we are dealing with mean

For the null hypothesis,

µ = 64

For the alternative hypothesis,

µ ≠ 64

This is a two tailed test.

Since the population standard deviation is given, z score would be determined from the normal distribution table. The formula is

z = (x - µ)/(σ/√n)

Where

x = sample mean

µ = population mean

σ = population standard deviation

n = number of samples

From the information given,

µ = 64

x = 65

σ = 3.5

n = 49

z = (65 - 64)/(3.5/√49) = 2

Recall, population mean is 64

The difference between sample sample mean and population mean is 65 - 64 = 1

Since the curve is symmetrical and it is a two tailed test, the x value for the left tail is 64 - 1 = 63

the x value for the right tail is 64 + 1 = 65

These means are higher and lower than the null mean. Thus, they are evidence in favour of the alternative hypothesis. We will look at the area in both tails. Since it is showing in one tail only, we would double the area. The probability value for the area above the z score from the normal distribution table is 1 - 0.97725 = 0.02275

We would double this area to include the area in the left tail of z = - 2. Thus

p = 0.02275 × 2 = 0.0455

Answer 2

`z=(65-64)/((3.5)/(√(49)))=2`

Now we can find the p value based in the alternative hypothesis and we got:

`p_v =2*P(z>2)=0.0455`

Explanation:

Information given

`\bar X=65` represent the sample mean

`\sigma=3.5` represent the population standard deviation

`n=49` sample size

`\mu_o =64` represent the value to verify

z would represent the statistic

`p_v` represent the p value

System of hypothesis

We want to verify if the true mean for this case is 64 or no, the system of hypothesis would be:

Null hypothesis:
`\mu = 64`

Alternative hypothesis:
`\mu \\eq 64`

The statistic is given by:

`z=(\bar X-\mu_o)/((\sigma)/(√(n)))` (1)

The statistic for this case is given by:

`z=(65-64)/((3.5)/(√(49)))=2`

Now we can find the p value based in the alternative hypothesis and we got:

`p_v =2*P(z>2)=0.0455`